by Elliot Benjamin, Ph.D. November, 2017
Introduction
As I traverse through my late 60’s, to my utter surprise I have found myself involved with a spiritual community, the Shambahala organization,1 which has.....]]>
by Elliot Benjamin, Ph.D. November, 2017
Introduction
As I traverse through my late 60’s, to my utter surprise I have found myself involved with a spiritual community, the Shambahala organization,^{1 }which has its lineage in Buddhist ancestry, but was initiated in the West by Chogyam Trungpa in the 1970’s.^{2} Now Shambhala and Trungpa is not in my Modern Religions book^{3 }only because I had never experienced this religion and my book is an experiential account of my experiences in modern religious movements. However, it was common knowledge for anyone familiar with religious cults that Trungpa was a controversial guru with a reputation for being both an alcoholic and engaging in numerous sexual encounters with his devotees.^{2 }But Trungpa died in 1987, and his son Sakyong Mipham, who has been the Shambhala guru since 1990, is apparently a bona fide ethical and upstanding human being in his personal life.^{4}
To backtrack a bit, my wife Dorothy had done a week-long Shambhala retreat in Nova Scotia a few years ago, and she completed her final Shambhala basic training level (Level 5) in November, 2016, which happened to be a few days after the U.S. presidential election. And I must say that I was struck by how “transformed” Dorothy was when she returned home, as we were initially both so devastated with the outcome of the election and our new U.S. President Trump. She came home conveying to me all about the vision of basic goodness in the world that she received from her weekend, symbolized as The Great Eastern Sun,^{5} and this had a rejuvenating effect upon me as well.
Although Dorothy had previously remarked that she would love for me to experience the joy of
being in a group of authentic spiritual people and that I might like Shambhala myself, I never took this
very seriously as I did not see myself wanting to meditate for an entire weekend with a group of people, and neither of us pursued the idea. However, after the presidential election, seeing how transformed Dorothy was when she returned home, I started to wonder if perhaps I might also get some benefit out of doing the Shambhala Level 1 training. Furthermore, I wanted to share Dorothy’s spiritual path with her, it it were at all possible for me to do so, and thus I decided to “jump” into the Shambhala Level 1 basic training weekend, in March, 2017, in Rockland, Maine.
It is now over nine months later, and I have completed the five basic Shambhala levels, as well as as the culminating Rigden level where one takes the Shambhala vow and receives a Shambhala name. Therefore I think it is high time for me to thoroughly examine what I have gotten myself involved in, and explore the issue of whether or not Shambhala is a cult. And I will do so in the same integrative (as opposed to “integral”) perspective context that I have used in a number of my previous Integral World essays^{6 }and conclude with an experiential cults analysis of Shambhala, relying primarily on the Bonewits Cult Danger Scale, which I utilized in my Modern Religions book.^{3} However, this essay in particular relies heavily upon my own experiences from being involved in Shambhala, and I therefore refer to my perspective here as “integrative experiential.”
I Jump—My Level 1 Shambhala Training (written in March, 2017)
While I was at my Level 1 training, I felt concerned about the disturbing aspects of Trungpa’s personal life, which I had learned a great deal about,^{2} but I was able to voice my concerns to both the director and one of the assistant directors at my Level 1 weekend, and I was treated with respect and taken seriously. The assistant director asked me, during my interview with her, if I would be willing to trade in the “history for the mystery.” This occurred soon after she asked me if I had any ethical concerns about the personal life of Trungpa’s son and current leader of Shambhala, Sakyong Mipham,^{4} and I honestly replied that I did not. Mipham has been the leader of Shambhala since 1990, and I have not seen anything controversial whatsoever about his personal life, which is rather amazing for how long he has been in this position of authority. The book that I read by Mipham, Turning the Mind into an Ally,^{7} was comforting to me, and I found nothing in it that I could object to. And perhaps this is the bottom line for me, and is consistent with Dorothy’s initial response to my concerns about Trungpa—as she said, “the son” is decent and ethical.
Now a part of me is rebelling in my new involvement in Shambahala, as I am wondering if I am being a hypocrite after all I have written about the dangers of modern religions,^{3 }and I am concerned that this will all backfire big time with Dorothy, as my feelings could easily get the best of me and I could start attacking Shambhala as a cult. But I am already signed up for Shambhala Level 2 next month, and the tentative plan is for me to do Shambhala Level 3 the month after that, Shambhala Levels 4 and 5 in the Fall, and then in December 2017 for both Dorothy and I to attend the post-levels training weekend in which we become formally accepted into the Shambhala community, take Shambhala vows, and actually receive Shambhala names.
But perhaps I should have my head examined. How in the world can I be doing all this? Me, the authentic philosopher and individualist, who is all too aware of the dangers of modern religions and cults. I cannot forget that Trungpa, in addition to having been an alcoholic, “sexaholic,” and possibly a heavy user of cocaine,^{2 } induced personally destructive behavior on at least one occasion, where a female participant, who attended one of his workshops with her husband, who was a well-known poet, was forcibly stripped naked against her will.^{2} Even more disturbing, it may to be the case that Trungpa knew about his successor Osel Tendzin’s having aides while he was sexually active with hundreds of devotees—both male and female—in the Shambhala community, and it was alleged by Tendzin that Trungpa reassured him that he could continue his sexual activities as long as he put spiritual intention into not transmitting his disease to anyone.^{8} Apparently a number of people contacted aides and died from their sexual contacts with Tendzin.^{8 }And Trungpa is the spiritual founder of the movement that I am now engaging myself in!
Well—what more can I say here? I had a good Shambhala weekend and I liked the actual
meditating as well as the one-on-one interactions and small group discussions with the other participants. The meditation was comforting and relaxing, and it was settling for me to get away from my computer for the weekend, with all my enmeshing political involvements since the election. I had lively and authentic conversations with the Level 1 director, who is quite the character, as he is a musician, therapist, and previously was an actor. Although I avoided having breakfast and lunch with the group, as well as the social chit chat during breaks,^{9} I chose to go to dinner with the director, both the assistant directors, a staff member, and four other participants. As it turned out, I ended up giving everyone a stimulating and humorous lesson about perfect numbers on a napkin,^{10 }and I had a great time, as the director and one other participant became keenly interested in my perfect number lesson. I achieved quite the reputation from this, and I felt like I became known in this way in an important part of my real self at my Level 1 Shambhala weekend.
My Level 2 Shambhala Training—My Good Feelngs About Shambhala Continue Amidst
My Continuing Concerns About the Shambhala Founder (written in May, 2017)
Dorothy ended up assisting at my Level 2 training, which was in Brunswick, Mane, and the training was rather pleasant for me, as I had worked through my conflicts about Trungpa at the Level 1 training, and I once again appreciated the peace and quiet and taking a break from my computer and political enmeshment. I didn’t resonate with the Level 2 director like I did at my Level 1 training, but the Level 2 director struck me as ethical and intelligent, though she was also quite serious and much more formal than the Level 1 director. In my Saturday interview with one of the assistant directors (who was the assistant director at my Level 1 training whom I did not have an interview with), I conveyed how the training was going well for me but that I missed the small group interactions and I thought people would speak up much more easily if this were done.^{11}
In between my Level 1 and Level 2 trainings, I had read Trungpa’s book Cutting Through Spiritual Materialism,^{12 }which was disappointing and concerning to me. Trungpa described in this book about the concept of “crazy wisdom”^{13} and conveyed alarming scenes of guru discipleship involving mental and physical abuse. He also described what appeared to me as nonsensical ramblings about mystical spiritual esoteric Buddhist beliefs, which were completely preposterous to me. Thus I had serious misgivings that this whole Shambhala mission I was putting myself through could easily backfire. But actually being in the training was relaxing for me, and although I did not feel comfortable with one or two of the participants, for the most part the participants seemed like good people to me, and I enjoyed seeing two participants from my Level 1 training, plus one of the assistant directors.
My Level 3 Shambhala Training: Still Feeling Good but I know My Limits With How Far
I Can Go With Shambhala (written in May, 2017)
I asked the co-director who I had my first interview with at the Level 3 training if Shambhala had expectations for participants to adopt the Buddhist spiritual and philosophical beliefs, in particular about reincarnation, as one proceeded to higher levels in the program, including the Rigden naming ceremony where one takes the Shambahla vow and receives a Shambhala name, after completing all five basic levels. And he reassured me that there were absolutely no expectations at these levels, though he honestly acknowledged that there were expectations about this at higher Shambhala levels. When I conveyed to him about the concerns I had in regard to Trungpa that I had talked about with the Level 1 director and assistant director, ^{2, 8 }he appeared to genuinely appreciate me bringing this up, and said that not enough people do this and it “keeps us honest.”
I stayed at the party/reception at the end of the program for a while, and I learned that this co-director had gotten his Ph.D in psychology through doing his dissertation on Acceptance and Commitment Therapy (ACT), which I had experienced in a weekend a number of years ago, had good feelings about, and had included in some of my articles about humanistic psychology.^{14 }I also learned that both his parents and his wife’s parents had been very involved with Shambhala and Trungpa in the 1970’s, which together with his Asian heritage, gave me a hint about why he placed a great deal of importance on ceremonial bowing between teacher and student,^{15} which he required me to do at the beginning and end of our interview, which was not particularly comfortable for me, but not a big deal either. And this co-director became more interested in me as I shared with him about my two Ph.D’s in mathematics and psychology, my writings, and my life after death experiential dissertation,^{16} and he said we should exchange contact information as he put my phone number and e-mail address in his computer device.
Although I know that it would backfire big time for me to go any further with Shambhala past the basic five levels and culminating Rigden training, I must give Shambhala a great deal of credit for being genuinely giving of their time to voluntarily help people without any financial compensation, and to be so open to work with people like myself, i.e., declared agnostics who appreciate the Shambhala meditative quiet and discipline but who are not willing to accept the spiritual Buddhist tenets of reincarnation, and who have concerns and misgivings about the original founder of Shambhala. I am eager to apply my cult danger analysis that I used in my Modern Religions book to Shambhala, which I will do after I complete all five levels and the Rigden “taking vows” training in December. Level 4 is not offered until September, three and a half months away, and I can use a respite.
Levels 4 and 5, Shambhala workshops with Dorothy, Rigden—Shambhala Vows and Name:
I’m a Shambhalian! (written in December, 2017)
So much has happened for me with Shambhala in the nearly seven months since I have last written about my Shambhala experiences. I’m going to cut to the chase here and just briefly say that I did my Shambhala Level 4 training in June and my Shambhala Level 5 training in September, and they were both beneficial and rewarding experiences for me. At both trainings I attended the Saturday night social dinners and actually had a good time, engaging in authentic conversations with participants, staff members, and the directors. I continued to voice my political concerns about President Trump in group discussions, and I felt the benefits of meditating to keep a semblance of balance and calmness in the midst of the devastating political turmoil that I felt continuously enmeshed in. I voiced my concerns about Trungpa to the quite elderly Level 5 director, who actually had personal contact with Trungpa in the 1970’s and knew Tendrin, Trungpa’s assistant who infected Shambhala devotees with aids (see above). The Level 5 director acknowledged some of my concerns, but when I conveyed about how Trungpa allegedly told Tendrin that if he kept his spiritual discipline intact that he didn’t have to worry about infecting people with aids when having sexual contact with them,^{8} the Level 5 director surprised me as he actually got upset and emotionally responded by saying “This was Tendrin’s bullshit!” It turns out the he also knew Tendrin, and had nothing good to say about him. And this put my mind to rest, as I was now willing to give Trungpa the benefit of the doubt on at least this issue.
As things progressed for me through Level 5, finally some ritual and hint of future further ritual expectations was introduced. We did some chanting, the Level 5 director talked about his leaving out food or wine for “sacrifices” as part of his meditation practice, and I learned that various Shambhala
centers engaged in chanting on a regular basis, even for newcomers. There was more talk about the upcoming Shambhala levels where the basic five levels are gone through from an intensive Buddhist Shambhala perspective of the religious theory involved, and I knew absolutely that this was something
that would backfire for me big-time if I ever tried to go any further with Shambhala. There was also quite the sales pitch given to us at the end of our Level 5 training, to become official Shambhala members, pay yearly dues, and volunteer to help out at various Shambhala functions, including assisting at the basic level trainings. And at the social dinner I attended during my Level 4 training, I learned about the very strange esoteric part of Shamhala that involved military-type training and uniforms, for the purpose of protecting Shambhala. But I had good feelings about all the co-directors of my Level 4 and Level 5 trainings, and one of the co-directors of my Level 5 training was the original Brunswick/Portland Shambhala founder. I especially appreciated my engagement with this co-director about my political enmeshment and concerns, both in my private interview with him and in the group discussion that he facilitated, as his own background was in government and politics, and he very much related to my communications about this.
And for the other part of my Shambhala involvement, in regard to my initial motivation of jumping into Shambhala to share a spiritual path with Dorothy, things were also going well. Dorothy and I did a half-day Shambhala workshop in Rockland in June, and it was a positive experience for both of us. This got reinforced as we attended another half-day Shambhala workshop in Rockland in September, where the original Brunswick/Portland Shambhala founder held a brief ceremony and gave us and a few other people our Shambhala pins for completing the five basic levels. The stage was set for us to do our grand finale, which was the Shambhala Rigden training December 1 – 3, in which we would take Shambhala vows and receive our Shambhala names. And we did this about three weeks ago.
Well Dorothy and I are both Shambhalians now. Our Rigden workshop (Rigden refers to the reputed ancient Shambhala spiritual kings^{17}) was in Brunswick and was basically a very good experience for both of us. The Rigden director was gay, from New York City, and quite outspoken about his lifestyle. He had a successful career in Wall Street. was apparently quite wealthy, and was an interesting character. As it turned out, he was very high up in the Shambhala organization, having been with Trungpa in the 1970’s and had recently co-taught in New York City with Trungpa’s son Mipham, and a number of visiting Shambhala higher-ups, including my Level 4 director, attended the Friday night session to hear him give his presentation. One particular Shambhala technique given to us by this director at the Rigden training, that I especially appreciated, was referred to as the Windhorse meditation, which involved a short meditation that lasted under a minute and could be used to prepare for events in the real world. Each of the previous five Shambhala training levels had been focused on a different kind of meditation and perspective, but for me I did not experience much difference in my meditation throughout the levels, as I just appreciated the quiet time to get in touch with my inner self. But the Windhorse meditation technique was something that I appreciated as “different,” and it was helpful for me to spend nearly 40 minutes practicing this by the lake, as part of our Shambhala Rigden exercises. It turns out that Dorothy also very much appreciated this exercise, as she had a powerful experience with it.
As the weekend continued, the focus became more and more on taking the Shamahala vow, and we were instructed that in our Saturday afternoon interview with the director, that we should formally request that we would like to take the Shambhala vow. This surprised me, as I had thought the taking of the Shambhala vow was optional, and Dorothy and I had discussed whether or not we would agree to take the Shambhala vow—thinking that we probably would but that it was not definite. However, it was helpful for me to know that the vow was essentially to work toward an “enlightened society,” seeing the potential of good in people, without any kind of esoteric religious expectations.
But soon before I would be having my Saturday afternoon interview with the director, as I read through all the specifics that I would be agreeing to in taking the Shambhala vow, I came across one phrase that disturbed me. It had to do with pledging my allegiance to the “Sakyong,” which is the title given to the head of Shambhala—originally it was Trungpa and now it is his son Mipham. Well all
my work on modern religions and cults^{3 }came back to me, as well as my concerns about Trungpa, and
I knew that even though I had no problems with Mipham, that I could not blindly pledge my allegiance
to any future leader of Shambhala just because he (or she) was designated as the Sakyong. I doubted if anyone else gave this much thought, but for me it was a problem. And thus my dilemma—after all I worked for throughout Shambhala to share this spiritual path with Dorothy, and now we were at the culmination of finally being able to do this, it could all be sabotaged by my not being willing or able to take the Shambhala vow. I would be the only one in our group of about 30 participants who refused to take the vow.
But to thine own self be true—and I had no choice. However, I did have a creative potential resolution, which came to me as I had a bit of time to continue meditating until my interview with the director. What I came to is that if I thought of giving my allegiance to the Sakyong as giving my allegiance to the “principles” of Shambhala, meaning working toward an enlightened society and seeing the potential good in people, then perhaps I could feel o.k about taking the Shambhala vow. In other words, I would convey to the director that I could take the Shambhala vow if I could interpret my allegiance to the Sakyong to be not allegiance to any particular person who happens to be in this role, but only to a person in this role if he or she is upholding the principles of Shambhala, which I felt fine about. And this is what I did—and it worked!
I surprised the director by saying instead of that I would like to request taking the Shambhala vow, that I would like to “discuss” taking the Shambhala vow. And he responded well to my interpretation of giving my allegiance to the Sakyong, actually quite well. I was careful to convey how I had no
problem with Mipham but was thinking of a possible future disturbing occurrence, even though I said that in all likelihood there was little danger of something disturbing like this ever occurring in Shambhala, and a bit about that I had a background in studying cults with unethical gurus. I could have
let things stand with his good response to this, but I knew I had to go further to make sure that I was truly being myself before taking the Shambhala vow, and so I also conveyed to him about my concerns about Trungpa, Trungpa’s allegedly legitimizing his assistant Tendrin’s deadly behavior in affecting people with aids, and how I did not consider some of Trungpa’s behaviors to be ethical or consistent with how I viewed the principles of Shambhala. As it turned out, Tendrin had been this director’s mentor as well as lover, and he had great admiration for Tendrin, defended him, minimized the extent of his destructiveness in infecting people with aids, and offered that he could think of alternative explanations for Trungpa’s violating behavior with ordering the poet’s wife to be stripped naked (see above),^{8} etc. Then he said we were out of time and I could ask him later about what he was thinking of for a possible alternative explanation for this. But we parted on good terms, and he thanked me for my honest and open questioning.
And there you have it. The Rigden director had a completely opposite perspective from the Level 5 director about Tendrin, who had told me that “this was just Tendrin’s bullshit,” and from my perspective the Rigden director had a blind side to some of the most egregious faults in the history of Shambhala. But he accepted my questioning and concerns about this, and this was what was most important to me. Both Dorothy and I subsequently took our Shambhala vows, with a number of guests attending to observe the ceremony, and we received our Shambhala names. And the director gave me the name of “Wisdom Diamond,” which was very special to me, as this made it clear how much he appreciated my questioning of Shambhala.
This is pretty much the end of the story for me. In the next section I will be examining how Shambhala fares on my cults scale that I have used with all my modern religious involvements,^{3 }and I am expecting that Shambhala will fare well. However, I will mention that one concern I had as the Riden weekend came to its close, was with the power and authority of the Rigen director. He decided on Shambhala names for all the participants based upon his quick interviews with them, and conveyed to us how he chose these names very carefully and that the names were very meaningful, and that if we did not right away appreciate their significance for us, that we may appreciate their significance in a few years. Although I like my name, I don’t think everyone necessarily felt the same way about their names. This does raise some concerns for me about the power and authority given to Shambhala higher-up directors, and especially the Sakyong, and my concerns about this will be included in my cults analysis in the next section.
However, my experience with the Rigden director being open and flexible with me extended to his respectful response to my publicly sharing that I was not comfortable with reciting one of the chants that we were instructed to chant during our Rigden taking vows ceremony. This chant had to do with paying homage to the ancient Shambhala warriors, and the language attested to their victories through making war on their enemies. When I asked if this referred to actual “war,” the Rigden director affirmed that it did, and defended the Shambhala warriors as fighting against the unethical powerful people in their society who were destroying their culture, and actually compared it to the situation with U.S. President Trump, as I had already been quite outspoken about my Trump concerns and he had acknowledged my concerns and apparently shared them. But without knowing the specifics of what justified these Shambhala warriors in engaging in war and killing people, I did not feel comfortable in reciting these vows that paid homage to them, which is what I conveyed publicly to the Rigden director. And once again the Rigden director listened to me respectfully, as he conveyed that if a vow was not comfortable for anyone then we should just “sit it out” and not feel any pressure to recite the
vow. And I felt that there was not much more that I could ask for in setting my mind at ease that I was not blindly following a dangerous Shambhala guru.
Analysis of Cult Dangers of Shambhala
It is now time to conduct my experiential analysis of the cult dangers of Shambhala. To do so, I will utilize the same three rating scales that I utilized in my Modern Religions book, though I will now primarily make use of the Bonewits Cult Danger Scale,^{18} which is the rating scale that I have found
to be the most useful in my analysis of the cult dangers of all the modern religious groups that I have previously experienced and analyzed.^{3 }The nuts and bolts of the Bonewits Cult Danger Scale are
as follows:
THE BONEWITS CULT DANGER SCALE
1. INTERNAL CONTROL: amount of internal political power
exercised by leader(s) over members.
2. WISDOM CLAIMED: by leader(s); amount of infallibility
declared about decisions.
3. WISDOM CREDITED: to leaders by members; amount of trust
in decisions made by leader(s).
4. DOGMA: rigidity of reality concepts taught; of amount of
doctrinal inflexibility.
5. RECRUITING: emphasis put on attracting new members, amount
of proselytizing.
6. FRONT GROUPS: number of subsidiary groups using a different
name from that of the main group.
7. WEALTH: amount of money and/or property desired or obtained;
emphasis on members’ donations.
8. POLITICAL POWER: amount of external political influence
desired or obtained.
9. SEXUAL MANIPULATION: of members by leader(s); amount
of control over the lives of members.
10. CENSORSHIP: amount of control over members’ access to
outside opinions on group, its doctrines or leader(s).
11. DROPOUT CONTROL: intensity of efforts directed at
preventing or returning dropouts.
12. ENDORSEMENT OF VIOLENCE: when used by or for the
group or its leader(s).
13. PARANOIA: amount of fear concerning real or imagined
enemies; perceived power of opponents.
14. GRIMNESS: amount of disapproval concerning jokes about the
group, its doctrines or leaders(s).
15. SURRENDER OF WILL: emphasis on members not having to be And here are
responsible for personal decisions.
RATING SCALE
LOW HIGH
1 2 3 4 5 6 7 8 9 10
And here are my Bonewits Cult Danger Scale ratings of Shambhala, along with my total and average
(rounded off to one decimal place) ratings:
Internal Control: 1
Wisdom Claimed: 9
Wisdom Credited: 9
Dogma: 9
Recruiting: 3
Front Groups: 2
Wealth: 3
Political Power: 1
Sexual Manipulation: 1
Censorship: 1
Dropout Control: 1
Endorsement of Violence: 1
Paranoia: 2
Grimness: 2
Surrender of Will: 2
Total: 47
Average: 3.1
The way that I came up with my above experiential Shambhala ratings is as follows. First off, I am quite comfortable with having given Shambhala the lowest (best) rating of “1”when it comes to Internal Control. I experienced absolutely no attempts by Shambhala to control any aspects of my personal life. On the other hand, I gave Shambhala very high (i.e., very alarming) ratings of “9” in the categories of Wisdom Claimed, Wisdom Credited, and Dogma. Yes there is a great deal of wisdom claimed by Shambhala’s original founder Trungpa, and to a lesser extent by his son Mipham, the current leader of Shambhala. Furthermore, the wisdom credited to the Shambhala leaders, especially Trungpa, is certainly very high, with their pictures in prominent display as a central feature of the Shambhala altar at the Brunswick center. However, it is also the case that I was able to openly voice my concerns about Trungpa to a number of the Shambhala directors at my levels trainings, and Mipham appears to be much less of a guru figure compared to his father. Thus I refrained from giving Shambhala the highest “10” ratings in these categories. Similarly, there is a great deal of dogmatic religious kind of belief in Shambhala, following essentially all the guiding principles, esoteric religious/spiritual beliefs, and training practices as originally set forth by Trungpa. However, I also found that I was able to voice my concerns about following some of these dogmatic practices, such as publicly conveying in my Rigden weekend to the high level director that I was not comfortable with reciting one of the chants that paid homage to the ancient Shambhala warriors, and it was accepted by this director without any problem. Therefore I have refrained from giving Shambhala the highest “10” rating in the Dogma category.
I must admit that with these early three very alarming ratings that I was giving Shambhala, I started to wonder if Shambhala would rate at a higher level of cult danger than I had anticipated, based upon the generally positive experiences I have had with Shambhala. But as I continued my analysis, the ratings I was giving to Shambhala significantly started to improve. One striking observation is that I gave Shambahla five more ratings of “1,” in the categories of political power, sexual manipulation, censorship, dropout control, and endorsement of violence. Furthermore, I gave Shambhala quite
favorable ratings of “2” in the categories of front groups, paranoia, grimness, and surrender of will. For a bit of explanation of why I gave these ratings as “2” instead of “1”: I had seen flyers for Shambhala talks about various topics without mentioning that these talks were sponsored by
Shambhala; I had received a rather strange invitation from a Shambhala assistant that she had read about my work on cults and was inviting me to give a talk to the Brunswick/Portland (Maine)
Shambhala board of directors about whether I considered Shambahla to be a cult (I decided to not
respond to this invitation); although most of my levels trainings directors balanced their deep Shambhala teachings with a good sense of humor and lightness, the director at my Level 2 training was quite serious virtually for the whole training weekend; and though I did not experience giving up my will to Shambhala per se, there is an aspect in Shambhala of giving up your will to the higher universe, in a spiritual context. However, these considerations were all minor for me, and I was quite comfortable with giving Shambahla my ratings of “2” in these categories.
For the remaining categories of Wealth and Recruiting, I gave Shambhala the rating of “3.” This signifies for me that I did not have any serious concerns, but that there was some degree of caution that I experienced in these categories. For Wealth, there is a definite emphasis in Shambhala on becoming members and paying a yearly membership fee that is not unreasonable, but also not negligible. There is also an emphasis on Shambhala members making higher donations to Shambhala if they are able to do so, for the purpose of giving scholarships to Shambhala aspirants who have financial difficulty with the costs of the workshops. If one advances in Shambhala, the higher level workshops, although not exorbitant, or also not “cheap,” and all things considered I therefore felt that Shambhala was pragmatic and self-sufficient in the category of wealth, warranting a rating of “3.” And in regard to Recruiting, Shambhala very actively conveys all their events in weekly internet communications to anyone on their mailing lists, with frequent separate promotions of some of their particular events. This is certainly not at a degree that is inappropriate, but it does illustrate what I consider to be reasonable and substantial
efforts at recruiting, and justifies my rating of “3” in this category.
It is now time to determine how these numbers fare in regard to Shambhala having possible cult dangers. In my Modern Religions book^{3} I determined five categories of cult dangers, inclusive of a Favorable category: High Cult Danger, Moderate Cult Danger, Minimal Cult Danger, Neutral, and Favorable.^{19 }Based upon my experiences and numerical ratings in the three cult danger scales that I had
utilized in my Modern Religions book, I placed Scientology and The Unification Church in the High Cult Danger category, and I included Neopaganaism and Kripalu Center for Yoga & Health in the Favorable category. In terms of their respective Bonewits Cult Danger Scale average ratings, I gave The Unification Church the highest rating of 9.0 and Neopaganism the lowest rating of 2.1. In regard to the Neutral category, I included A Course in Miracles with a rating of 3.5. It thus appears, from the above Bonewits Cult Danger Scale rating of 3.1 that I gave to Shambhala, that we have an organization here that may fall somewhere in between Neutral and Favorable in regard to its cult dangers. More specifically, the lowest numerical rating I gave to an organization that I classified as Neutral was to the International Cultic Studies Association, with a rating of 3.4, and the highest numerical rating I gave to an organization that I classified as Favorable was to Acceptance & Commitment Therapy (ACT), with a rating of 2.7.^{3 }Thus with Shambhala’s rating of 3.1, it may make sense to create a new category for Shambhala that can be called “Mildly Beneficial.”
To justify further putting Shambhala in a Mildly Beneficial category, we can make use of the other two rating scales that I used in my Modern Religions book, which were the Anthony Typology and the Wilber Integral Model.^{3 }Without going into detail about the specifics of the formulations of these two scales (see [3] for more information about this), suffice it to say that in the Wilber Integral Model
I would place Shambhala in the Transrational category, which reflects Shambhala’s strong emphasis on making use of one’s mind in a variety of ways, but most importantly, transcending one’s mind through a disciplined practice of meditation. In the Anthony Typology I would classify Shambhala as being Monistic, meaning that all individuals are inherently good spiritual beings, and as being Multilevel, meaning engaging in genuine inner spiritual development. In regard to the Anthony Typology of Technical vs. Charismatic, I would place Shambhala pretty much in between the two, and in regard to the Wilber Integral Model of Legitimacy anchored in a tradition, I would also place Shambhala in the middle, as although there is a reputed long heritage of Shambhala (Rigden) kings over a period of
thousands of years,^{17} the Western form of Shambhala is very much one that was created by Trungpa in the 1970’s. In regard to the Phase Specific category in the Wilber Integral Model, I would give Shambhala a low rating, as the Shambhala guru (initially Trungpa and now his son Mipham) is a lifelong position, not at all phase specific. Finally, in regard to Wilber’s Four Quadrant Integral Model, I would give Shambhala high ratings, as although there is a dominant focus on one’s inner experience through meditation, there is also a focus on one’s body postures, breathing, physical exercises, and yoga, as well as communicating with others in dyads and small groups, and working in the broad social world to create an “enlightened society.” Thus incorporating the high ratings I give Shambhala for being Transrational, Monistic, Multilevel, and being supportive of the Wilber Four Quadrant Integral Model, I am comfortable placing Shambhala in the Mildly Beneficial category.
Conclusion
I never anticipated that I would embark on another “modern religion” as I traverse through my late 60’s. But as I have described above, my motivation to do so was extremely high, and I therefore “jumped” into Shambhala, and I am very thankful that I have apparently landed with both feet on the ground. Yes there are concerns to watch out for in regard to possible cult dangers for Shambhala, in particular regarding their strong reliance on religious dogma and paying homage to their gurus. However, I believe that the decency and open responsiveness of the Shambhala high level directors has thus far safeguarded Shambhala from entering into any kind of dangerous cult activities. The fact that I was able to voice all my concerns and criticisms to a number of these directors speaks volumes to me about why I do not consider Shambhala to exhibit serious cult dangers. Of course, it is possible that this situation could change in the future, in particular through the changing of the guard in the leadership of Shambhala, once Mipham dies and one of his daughters takes over his role as Shambhala guru. However, I do not think that that the current “benevolent” leadership in Shambhala is likely to change for the worse. Furthermore, from what I have experienced in all my Shambhala trainings this year, I must say that both my Shambhala meditation experiences as well as social experiences have been beneficial to me. In this world of overwhelmingly disturbing everyday events, it is very settling for me to have a spiritual practice where I can just be by myself and feel content. And having a spiritual group that I can relate to in this way and who is supportive of my practice, is something that I have learned to value in Shambhala. And of course most important of all to me, I am now sharing this spiritual path with my wife Dorothy. So in conclusion, at this time I am willing accept Shambhala as a bona fide spiritual community without showing any serious signs of cult dangers.
Notes and References
See the Shambhala organization international website at https://shambhala.org/about-shambhala/shambhala-global-community
For information about the life, teachings, and controversies for Trungpa see https://en.wikipedia.org/wiki/Chögyam_Trungpa
See Elliot Benjamin (2013), Modern Religions: An Experiential Analysis and Exposé. Winterport, Maine: Natural Dimension Publications.
For information about the life and teachings for Mipham see https://en.wikipedia.org/wiki/Sakyong_Mipham Note that are no controversies included in the informaton about Mipham given in Wikipedia, and in a thorough internet search I have not found any indications of unethical behavior for Mipham during his whole 28 year tenure as the
leader/guru of Shambhala.
In Shambhala teachings,The Great Eastern Sun represents the alleged force of basic goodness in the universe, and this is described as a fundamental premise of Shambhala in Trungpa’s (1984) book Shambhala: The Sacred Path of the Warrior. Boston: Shambhala Publications. This concept of basic goodness in Shambhala is also the dominant focus of the Shambhala Level 1 training.
See Elliot Benjamin: Integral vs. Integrative: A Response to Scott Parker (2007); Clinton vs. Trump: A More Integrated Perspective (2016); Trump vs. Hitler: an Integrative Perspective (2016); Fighting Against the Trump Dictatorship: An Integrative Perspective (2017); Resisting Trump: 10 Months Later from a More Narrow but More Honest Integrative Perspective (2017); at www.integralworld.net
See Sakyong Mipham (2003), Turning the Mind into an Ally. New York: Riverhead Books.
For information about Osel Tendrin in relation to his affecting people with aids, and the allegations regarding the related advice he received from Trungpa, see http://tibetanaltar.blogspot.com/2010/05/tibetan-roulette.html
I have a dominant introspective and introverted nature, which I have described in my book
The Creative Artist, Mental Disturbance, and Mental Health (2017). Winterport, Maine: Natural
Dimension Publications.
For information about perfect numbers, See Elliot Benjamin (2017), Numberama: Recreational Number Theory in the School System. Potomac, MD: Bentham Science; and Elliot Benjamin (2006), Integral Mathematics: A Four Quadrant Approach at www.integralworld.net
As it turned out, there was a small group discussion the next day, which was very impactful for me, relating to the “cocoon” people surround themselves in, which was a dominant theme in the Level 2 training.
See Chogyam Trungpa (1973), Cutting Through Spiritual Materialism. Boston: Shambhala
Publications.
See www.en.wikipedia.org/wiki/Divine_madness for a description of crazy wisdom in religious traditions. See a number of articles on the Integral World site at www.integralworld.net on the topic of the dangers of crazy wisdom as applied by gurus Adi Da and Andrew Cohen. See also Geoffrey Falk (2009), Stripping the Gurus: Sex, Violence, Abuse and Enlightenment. Million Monkeys Press, for a scathing and alarming exposé about the dangers of unethical gurus using crazy wisdom in a multitude of religions—both ancient and contemporary.
See in particular, Elliot Benjamin (2007), An Integrative/Non-Integral Psychotherapy Model at www.integralworld.net
As I continued to progress through my Shambhala trainings, ceremonial bowing took place more and more, as we bowed whenever we entered and left the meditation room/sanctuary and before and after each of our interviews with the directors. However, I was able to take this ceremonial bowing in stride, as I looked at it as a show of respect for the principles of Shambhala that I felt comfortable with, and not as any kind of homage to Shambhala gurus.
See Elliot Benjamin (2012), An Experiential Exploration of the Possibility of Life After Death Through the Ostensible Communications of Mediums with Deceased Persons. Saybrook University: UMI 3509443.
See https://en.wikipedia.org/wiki/Kings_of_Shambhala for description of the legend of the Shambhala Rigden kings, as well as Trungpa’s book The Sacred Path of the Warrior (see book information in [5]).
See Issac Bonewits (1989), Real Magic. York Beach, Maine: Samuel Weisner. (Original work published 1971)
Although it has been pointed out to me that it would have been more accurate to have weighted the various components of the Bonewits Cult Danger Scale that I have utilized, as obviously some of these components are more significant than others, I am giving them equal weight in my analysis of Shambhala in order to reasonably make a comparison of my cult danger ratings of Shambhala with those of the other religious organizations that I have used this scale for previously.
is taking place at the Belfast, Maine library on Monday 3/14 from
6:30 – 8:00 p.m. This is a monthly support/discussion group for
people interested in developing their creative artistic potentials.
Find like-minded creative spirits and camaraderie in overcoming
the obstacles to become a successful creative artist.
]]>is taking place at the Belfast, Maine library on Monday 3/14 from
6:30 – 8:00 p.m. This is a monthly support/discussion group for
people interested in developing their creative artistic potentials.
Find like-minded creative spirits and camaraderie in overcoming
the obstacles to become a successful creative artist.
]]>expose´. Raleigh, NC: Lulu.com. 377 pp. 978-1-257-08261-2. Paperback,
$18.44. Reviewed by Nori Muster.
Elliot Benjamin, Ph.D., describes his experiences in a variety of new age
spiritual organizations, most of which are psychology-based groups. He
describes each group and offers his ratings based on three academic scales in
use since the 1970s: the Anthony Typology, the Wilber Integral Model, and the
Bonewits.....]]>
expose´. Raleigh, NC: Lulu.com. 377 pp. 978-1-257-08261-2. Paperback,
$18.44. Reviewed by Nori Muster.
Elliot Benjamin, Ph.D., describes his experiences in a variety of new age
spiritual organizations, most of which are psychology-based groups. He
describes each group and offers his ratings based on three academic scales in
use since the 1970s: the Anthony Typology, the Wilber Integral Model, and the
Bonewits Cult Danger Scale. He then places the groups on a spectrum that
ranges from favorable and benign to high cult danger. The first hundred pages
of the book familiarize the reader with the scales and Benjamin’s method of
rating.
Developing a reliable rating method is useful, since it emphasizes the differences
between groups, and would prevent journalists and casual researchers from
lumping all new age spiritual organizations in the same category of danger.
Along with the more notorious groups, Benjamin rates about a dozen groups
that he considers benign. This may help researchers who study group dynamics
to recognize what makes a group dangerous. It may also inform religious leaders
who want to fall on the favorable-benign side of the spectrum.
Researchers will find plentiful information on lesser known groups. Since
many of the groups described in the book are small, or not considered
dangerous, until now they may have been ignored in the cultic studies
literature. Hopefully, the information on benign groups will put some people’s
minds at ease. As the director of Steamboats.com, a website dedicated to
historic preservation, I once received a letter from a concerned mother
questioning her son’s employment as a deckhand on the Delta Queen
Steamboat. I assured her that it was most likely a positive experience for her
son that would look good on his resume´. Benjamin’s descriptions may bring
similar peace of mind to friends and relatives of people who dabble in the
benign groups he covers.
The Anthony Typology, developed by Dick Anthony, analyzes a group on the
scope of its beliefs, whether it is charismatic, and whether it is antagonistic
toward the outside world. The Wilber Integral Model, developed by Ken
Wilber, rates a group according to how controlling it is, and whether its
philosophy has a rational or traditional basis. The Bonewits scale, developed
by Issac Bonewits, assigns a number between one (low danger) and ten (high
danger) on fifteen traits, such as the leader’s claim of wisdom, the amount of
wisdom attributed (blind followers), and rigidity of dogma. Bonewits rates how
much a group is interested in money and political power; as well as the
common hallmarks of a dangerous cult: sexual abuse, censorship, endorsement
of violence, paranoia, lack of sense of humor about itself, internal control of
members, and surrender of will. The ratings are added up and divided by
fifteen to come up with an average cult danger rating.
Benjamin describes each group, then rates each on the three scales, and follows
with his rationalization for why he rated each group as he did. He admits that
his ratings are purely subjective, based on his experiences. Individual
researchers will certainly disagree with some of Benjamin’s ratings, and
certainly the groups themselves will disagree if they have a bad rating.
106 The Journal of Transpersonal Psychology, 2012, Vol. 44, No. 1
One of the weaknesses of rating organizations is that it is difficult to see what is
going on behind the scenes. A researcher would have to stumble into the inner
circle of any group to find out what is really going on. Therefore, there is a
danger of falsely giving a group a benign rating. Even a homeowners’
association or bridge club may have the potential to inflict extreme emotional,
financial, or other abuses, which a casual observer may not notice. Also, it
must be kept in mind that groups can change. They may reform themselves or
turn sinister, based on who is in the group, and whether the system is ripe for
abuse, or ready for healing. In addition, once a group has been stained by
sexual or other violent forms of abuse, it may have a difficult time getting its
reputation back. Therefore, high ratings on the Sexual Manipulation and
Endorsement of Violence scales need to be more heavily weighted to get an
accurate picture of a group’s overall danger rating.
Another note is that it would be a mistake to apply the Bonewits scale to
political groups, as Benjamin has in essays outside of this volume. All political
groups would score high on several of the scales, such as Wisdom Claimed,
Wisdom Credited, and Dogma, and certainly they would score high on the
Wealth and Political Power scales. Since these five scales would be elevated, it
would be unfair to compare the average of a political group’s rating to the
average of a new age spiritual organization. To obtain a more accurate rating
of political organizations, a researcher would need to remove those five items,
and add five items to rate the group’s integrity. Does the group lie for political
gain? This would say more about whether a political group is dangerous than if
they want wealth. Needing money is built into the game of politics these days.
In Chapters Two and Three, Benjamin presents a collection of essays he wrote at
the time he was going through his group encounters. The essays are presented in
two sections, first the late 1990s and early 2000s; then the 1970s. Benjamin took
about fifteen years off in between to earn his Ph.D. in mathematics and establish
himself as a college professor with a specialization in pure mathematics. He
describes his academic pursuits as part of his spiritual search, since he spent years
working on pure mathematics for several hours each morning as a meditation.
Benjamin’s essays in Chapters Two and Three read like journal entries, written in
the moment. Many of these entries begin when he is enamored with a new group
he is exploring, then in a subsequent entry, he denounces the group and explains
what he dislikes about it. He seems to have a cast iron stomach for unusual group
experiences. Many ex-cult members and researchers may experience the gack
factor (feeing repulsed) by some of Benjamin’s realizations as a naı¨ve follower.
As an ex-member of an Eastern guru group, I have avoided all new age
religious organizations except a very few. The Philosophical Research Society,
founded by Manly Hall in 1934, was a short walk from where I lived in Los
Angeles in the early 2000s. I attended many lectures, workshops, and even a tai
chi class there with no adverse reactions. However, once in the late 1990s, I
attended a house party put on by members of a group Benjamin would rate as
mild. At one point, they got everyone’s attention to do a group meditation.
Everyone joined hands in a circle in the living room. This was an unbearable
Book Reviews 107
trigger for me and I waited out in the front yard until the ceremony was over.
In my experience, I would have found most of the situations Benjamin lived
through as undesirable for myself.
Benjamin describes his deepest and most conflicted affiliation in Chapter Four:
Encounters with Scientology. In a series of his characteristic journal-like
essays, he reveals little-known details about the group, such as how they get
people to join and what goes on in an auditing session. As a researcher, I never
knew much about Scientology before, but the book has given me a substantial
education on the group’s inner workings. Since Scientology is a highly secretive
group, I believe this is one of the book’s greatest contributions.
One of the most terrifying aspects of Benjamin’s experience was the amount of
money he invested in the various groups he joined. His non-cult friends and
family must have found themselves exasperated trying to prevent him from
wasting yet more of his hard-earned money chasing the next great thing.
Benjamin repeats a similar pattern in each group: He becomes intrigued, gets
hoodwinked for a sum of money, becomes disenchanted, and leaves. He
discusses the financial hardships of group involvement quite extensively, which
will be informative for seekers who are considering a similar path.
One would think that interest in joining coercive organizations would have
died down by now, hitting its peak in the 1970s. However, due to millennial
fears and economic hardship, more people than ever are attracted to bad
leaders. This book issues a warning that is needed now more than ever, and will
therefore appeal to scholars, as well as families and others who lose a loved one
to such groups.
Publishing this book is a milestone for Benjamin, since it is the culmination of
his nearly forty years of writing about alternative spiritual organizations. In
essence, he is an unapologetic cult-hopper, revealing in Chapter Five his
disappointment with the Jewish religion of his ancestors and the loss of his
father at the age of two as factors that may have led him to search for meaning
through new age group involvement. He also admits that he joined particular
groups after falling in love with women involved in the groups.
After describing and rating all of his group experiences, the book seems to
point to the need for a creative non-fiction rendering. It would be refreshing
to read a memoir by him that goes in chronological order, offering selected
scenes from his journey. He has already told us what he really thinks. Now all
that is left to do is to show us the worlds he has discovered sans any further
analysis.
REFERENCE
MUSTER, N. (1997). Betrayal of the spirit: My life behind the headlines of the Hare
Krishna movement. Champaign, IL: University of Illinois Press.
108 The Journal of Transpersonal Psychology, 2012, Vol. 44, No. 1
The Author
Elliot Benjamin, Ph.D., is a philosopher, mathematician, musician, counselor,
and writer, with Ph.D.’s in both mathematics and psychology. He has a
specialization in Consciousness and Spirituality, and is the author of over
seventy-five published articles in the fields of humanistic and transpersonal
psychology, parapsychology, spirituality and the awareness of cult dangers, art
and mental disturbance, pure mathematics, and mathematics enrichment. He
has been a mathematics professor for twenty-one years at Unity College in
Maine, and currently teaches mathematics and psychology at a number of
different on-campus, and online schools. He enjoys playing the piano and
tennis, and ballroom dancing.
The Reviewer
Nori Muster, M.S., is an author living in Arizona. She loves sharing stories
about her life and visiting with friends. Her first book, Betrayal of the Spirit
(1997), is her memoir of ten years in the International Society for Krishna
Consciousness (ISKCON). Former ISKCON members worldwide have
accepted her book as a mind-opening narrative of a critical decade in the
group’s history. She has written nine books that promote rational thinking and
recovery from systemic abuse, which are available on Amazon.com.
* * * * * * *
Book Reviews 109
]]>I had not intended to write any more essays about Andrew Cohen, as I thought I had put closure on my thoughts about whether Cohen’s May, 2015 apology was “truly sincere, manipulative, or other” [1]. However, about.....]]>
I had not intended to write any more essays about Andrew Cohen, as I thought I had put closure on my thoughts about whether Cohen’s May, 2015 apology was “truly sincere, manipulative, or other” [1]. However, about a month ago I received a request from William Yenner [2] that one of Cohen’s ex-students, who recently wrote a book about her experiences with Cohen, would like to get into correspondence with me, as she appreciated my Integral World essays about Andrew Cohen. I didn’t think too much about this, but soon after I agreed to the request, I became acquainted with Marlowe Sand (a pseudonym), who sent me her book: Paradise and Promises: Chronicles of My Life with a Self-Declared, Modern-day Buddha [3] in the hopes that I would write a review of her book. To be candid, writing reviews of books is not something I generally do per se, unless it is part of an essay that I have other reasons to be writing. But mostly out of respect for William Yenner, I replied that I would be willing to read her book and was open to writing something, but that it would probably take me a long time to do so. About 2 weeks ago I started reading Marlowe’s book, just for the hell of it. And to my surprise, the Prologue captivated me right away, as I learned that Marlowe was one of those survivors of the “freezing lake” incident (cf. [2], [3]), and her portrayal was poignant and moving to me:
We are committed to prostrating ourselves in the water for an hour without stopping….Two weeks previously, in a group of 25 women, we had been the ones who staggered out of the water before the hour was up….We stand four feet apart, waist deep in the water. At first, each plunge shocks my whole body, and I dread going under the second, third, and fourth time. After ten minutes, I am so desperate that it is inconceivable that I could still be here in 50 minutes. My whole body is shaking convulsively, teeth rattling uncontrollably—banging, not chattering. “Face everything and avoid nothing!” After 30 minutes, my face shrinks tight against my skull. My brain feels as if it has shriveled inside my head. I can’t feel any part of my body. I am afraid. (cf. [3], p. 5)
I knew right away that I would be reading Marlowe’s book much sooner than I had anticipated, and I conveyed this to Marlowe and asked her again to convey to me her response to Cohen’s 2015 apology letter. Marlowe was now much more responsive to my request than she had previously been, and quickly sent me her insightful detailed responses to much of what Cohen wrote, with permission for me to use them in an Integral World essay. Given the intense and disturbing life-changing experiences that Marlowe had as a student of Cohen for 15 years, I think her informative responses to Cohen’s apology deserve to be publicly known. However, before conveying her responses to Cohen’s apology, I believe it would be relevant and informative to describe Marlowe’s thoughts and assimilation of her experiences, 5 years after leaving the movement, during a reunion with ex-community members in Costa Rica:
He would laugh raucously during stories of his students’ shock and pain. His cruelty, even sadism, had been relentless for most of his years of teaching….I renewed connection with a woman I had not seen in 25 years. She told me that while her daughter was dying she was systematically intimidated and attacked for not being sufficiently committed to the teachings…The friend of a woman who died slowly with her heart broken, after Andrew rejected her as a student, described how, 11 days before her death, a senior student berated her for 45 minutes for her selfishness, telling her that she would “die a miserable old woman.”….Women who wanted children sometimes had to choose between motherhood and being his student. The result was that some women felt pressured into having tubal ligations and one woman had an abortion….I had always imagined that Andrew was unaware of some of the cruelty meted out by his students. Now I learned from those in his inner circle that he had not only ordered it, but received detailed reports about it. I realized then that his condition was not just a mixed bag of benevolent and cruel; his was, or had become, a pathological condition. Nothing stood in the way of Andrew’s demands for control and adoration. There was no check on his abuse of power. (cf. [3], pp. 232-234)
However, Marlowe demonstrates the grace and open-mindedness of one who is able to ponder the complexity of human frailty mixed with the possibility of human spiritual attainment:
For most of 15 years, I was treated as one of the underclass. I was told on a daily basis that I didn’t have much potential. I bought into Andrew’s opinion that I was not up to much. Hell! I believed it for 15 years….We struggled to understand the nature of Andrew’s gifts and his terrible flaws. How can one separate his genuine spirituality from the base human motives which he so often indulged? He had claimed to teach love but was a bully. He taught surrender in an atmosphere of fear; vulnerability in the context of intense competition. He claimed to teach impersonally, yet cultivated favorites in an atmosphere of suspicion. Worst of all were his responses to budding spirituality among his students. In the name of rooting out ego, he crushed the spiritual impulse just as we began to trust our deepest selves. Was there something in him, we wondered, which made him incapable of empathy?….How do we understand that some of our experiences brought about greater awareness, and some nearly shattered some of us? Could it be that some remarkable things happened in spite of, or sometimes because of, a dangerous and destructive context? (pp. 237-239)
The alarming accounts of Andrew Cohen’s many years of being a destructive and sadistic guru to thousands of his followers, in the guise of nurturing their “spiritual development,” is well-documented in a number of books and articles by his ex-students and others, including articles on Integral World [4]. However, it is difficult and puzzling to know what to think about his 2015 second apology, which is much more comprehensive and detailed than his 2013 original apology, which could be dismissed much more easily as lacking depth and sincerity [5]. Let’s now see some of the informative and detailed responses to Andrew Cohen’s 2015 apology by Marlowe Sand. What follows (Marlowe Sand, 1/9/16, personal communication) are most of the statements given by Cohen in his apology, followed by excerpts from what Marlowe has sent me in response to these statements—see especially Marlowe Sand’s illuminating final summary comment.
(Cohen): During those years just the notion of higher development, the extraordinary possibility of emergence, would make my heart beat a little faster. It really WAS possible…and I could always feel the immanence of the miraculous always just around the corner. Over the years I took many risks so that great leaps forward could actually happen. I also whole-heartedly encouraged others, my students, to do the same….and so many of my students saw and felt the power and potential of what we had all given so much for. It was so exciting and such a grand spiritual adventure the likes of which most people never experience or even imagine.
(Sand): Tell that to some of the people whose loved ones were dying while he was humiliating them and downgrading them for their lack of commitment! For some it was a grand spiritual adventure, for some it was utterly degrading and destructive, for others (myself included) it was some kind of nightmarish mixture of the two. Any real reckoning with the past has to include an understanding of all of these things.
(Cohen): I gradually lost sight of people’s humanity, including my own, and only saw all of us as the living Self Aware consciousness that, in an evolutionary context, was going somewhere. And that was all that I believed was important or really mattered. I even stated this clearly and unequivocally at times when I was teaching.
(Sand): Yes he did.
(Cohen): And I was losing touch with my own simple humanity and everyone else’s. I also was simultaneously not paying attention to the gradual growing of my spiritual ambition, of my spiritual ego. I believe that my intense longing for the evolution of consciousness in my students was real, but I have begun to see more and more clearly how over time my pride and my desire for fame and recognition slowly but surely began to blur and corrupt my vision.
(Sand): Yes, here is getting to something real that he needs to look into.
(Cohen): The worst part of it is that I was oblivious to the many different ways some of my students were being pushed too hard and at times too relentlessly to make breakthroughs and too often breaking down as a result.
(Sand): There is something important missing here. His assumption is that he was consistently pushing his students towards truth but just pushed too hard. This is in need of serious reconsideration. In my opinion he was sometimes doing that but over the years more and more often he was just acting out his own frustration and unfettered aggression in increasingly random and extreme ways.
(Cohen): The very human, frail, fallible and vulnerable dimensions of myself that I was denying, I was simultaneously denying in those who had come to me for liberation. I was blind and ambitious and yet sincere in my spiritual aspirations as a teacher and as a thought leader.
(Sand): He later in the letter dismisses the possibility that he might also have been acting out of spite, sadism or cruelty. In retrospect, his raucous laughter when humiliating others looked more like spite than anything else. What does he mean when he insists that he was sincere? Whatever he means, he probably needs to get around to questioning whether he was!
(Cohen): In that movement from glorious experience to action one can make terrible mistakes. And as a thinker, I was moving and was still often creative in finding ever-new ways to express the inexpressible. And I was still curious.
(Sand): Curious about what? Not it would appear, about the inner experience of his students, nor about their individual development. He may have had a rigid idea about aiming for a specific style of collective expression and everything else including extreme suffering was subordinated to that idea. In fact even though he was very perceptive, and intuitive about people and their motivation, he may have been limited in really being able to relate to, or imagine or anticipate the consequences that his actions wrought. Certainly, there was something missing in his ability to empathize with others.
(Cohen): Even after 28 years of being a guide and a guru and a public thinker, I was still reaching and stretching to understand more and more about Life, Reality and the meaning/purpose of it all.
(Sand): Not stretching to understand how his behavior actually impacted people in the real world. I do believe that his excitement level may have continued but more and more in an ivory tower at one step removed form real people, with less and less reality checking from those around him (until those bold senior students in the last couple of years).
(Cohen): This fact of my still evolving and developing as a teacher made it that much easier for me to avoid and deny that slowly the world that I had given so much to give rise to over so many years, was beginning to crumble from the inside. My closest and most devoted senior students were beginning to see through my facade, could see that I was out of control, and see that I didn’t even know it. What made matters much worse is that I ignored the evidence; I ignored their respectful pleas for me to slow down and listen to them. For over six months during this period I literally couldn’t sleep, and night after night I convinced myself that I had no idea why this was the case. My self became more and more divided. I was still an inspired teacher and speaker, but I adamantly remained steadfastly and obstinately oblivious to the growing storm I was creating.
(Sand): Here we see Andrew Cohen beginning to feel what he did to a few of his close senior students at the end. I do not yet feel his sorrow for the other several thousand of us!
(Cohen): In those historic moments it all seemed worth it. But there were and have been too many moments where I simply have been wrong. Not only did my arrow miss the target but it caused unnecessary pain and suffering to too many people. For this I am deeply and terribly sorry. Too much suffering has resulted from my at times misguided efforts to create breakthroughs. I should have known better.
(Sand): He says the words, and this is good. But there is not any detail that shows us that he really has an example of having hurt someone in his mind and that he cares ABOUT THEM.
(Cohen): So many of you trusted me with your souls and I proved myself at certain pivotal moments unworthy of that trust. Again I am sorry. What I feel dreadful about is that the very idealism that I inspired and released in so many of you, I have wounded in the worst way possible. It’s difficult to bear that this is the case, but it just is.
(Sand): This paragraph strikes me as the real thing. I am glad he gets this. I hope at a deep enough level. His destruction of the spiritual impulse in some of his students is indeed a terrible thing.
(Cohen): I am committed to finding a way to honor all that was real and true that we stood for, for so many years. There is nothing else for me to do. There is nothing else I want to do.
(Sand): That is not enough. That would be a recipe for another, quite likely similar, chapter with a more benign face.
(Cohen): Eros is the VERTICAL manifestation of the Absolute principle. Agape is the HORIZONTAL manifestation of the Absolute principle. To say I neglected Agape is an understatement to be sure. Eros and Agape BOTH are essential ingredients of a truly Evolutionary Dharma. They BALANCE each other. They hold each other in a dynamic embrace of loving, creative and Integral tension. My over-emphasis on Eros with little respect for Agape created the circumstance where a collapse was inevitable. And that’s why it happened to fast…and for this I am to blame.
(Sand): Please Andrew Cohen! Drop the jargon and talk about some of the terrible things you actually did to people. If you want to generalize then use normal vocabulary in the English language—it is rich enough.
(Cohen): The deep pain. That’s what has made it possible for me to begin to truly let in the damage I have wrought and the harm I have caused to too many of you. I only wish I had been more awake to and in touch with my own flawed humanity form the very beginning.
(Sand): Again, the gap we feel is his being intimately in touch with another human. Was this hard for him before he was enlightened? Was this difficulty compounded by his guru role? This then was a terrible tragedy for Andrew Cohen. What did we do to him?
(Cohen): I often wonder how much of the outrageous evolutionary Fire could have awakened and been shared between us in the way that it was, without there being some kind of fallout, some measure of pain and suffering. And if that’s possible then how much would have been acceptable, and when would it all have become too much? At this point I really don’t know.
(Sand): So here he is bargaining. He asks, maybe to have the good stuff we had to have some of the bad stuff too? This is hair’s breadth away from saying, maybe it was worth it. Ask some of those on the A list if they thought it was worth it.
(Cohen): I do know that without the ultimate challenge this enormous calamity has given to me personally on a soul level, my own ego would never have backed down. It’s been extremely challenging on many levels to even begin to let in what has actually happened and why it has happened. And I know there is further to go.
(Sand): Good, I am glad he said that.
(Cohen): I am beginning to become simply human after so many years of hiding out in transcendence.
(Sand): Good. That is true.
(Cohen): It’s like coming back to earth after almost a quarter of a century of flying above the clouds. As much as I spoke about the need to “embrace heaven and earth,” I was obviously still rejecting so much of what it means to be a fully human being.
(Sand): Good. So Andrew might question many of his above assumptions in the light of this paragraph.
(Cohen): In so many ways I thought I was awake when I was clearly not.
(Sand): Good.
(Cohen): In my rejection of Agape, I was also rejecting the feminine principle in myself and in others and most painfully in women as a whole.
(Sand): Good. And much more for him to see here.
(Cohen): I am ashamed of how badly I blamed women for their evolutionary challenges.
(Sand): Here he appears to be holding on to his view that women are indeed particularly handicapped when it comes to enlightenment.
(Cohen): Many people accuse me of hating women. This is not and has never been true. But I was in so many ways arrogant and insensitive and even cruel in my impatience at times.
(Sand): Is he claiming that his insensitivity and cruelty were really just impatience and therefore not so bad?
(Cohen): I apologize to the women who were affected and am so very sorry for being so lacking in the real heart that was desperately needed.
(Sand): Good. This feels real.
(Cohen): I failed many of you in the worst way and for this I really have no excuse. I became a caricature of the very behavior and attitudes in men that I was so sure that I had transcended.
(Sand): Good.
(Cohen): And the painful and ironic truth in all of this is that I did have a real passion and commitment for a very radical expression of women’s liberation. I had seen a truly miraculous potential and possibility. But, in the end, I proved to have neither the patience, nor the skill, nor the deep humility and care (agape) to create the conditions that would have made a stable breakthrough actually possible.
(Sand): Good. Well said. But he still assumes that he SAW things clearly. Given that he now admits that the shadow was there yet unacknowledged, then all of his perceptions may have been less clear.
(Cohen): In the middle years of my teaching career, at times I came up with and tried many outrageous stunts in order to once again catalyze big breakthroughs. Also to be honest I was many times actually in a state of desperation because I cared so much, and was trying to get my students to care as much as I did about what was possible, the very promise we had all given our lives for.
(Sand): It is a big assumption that he did what he did because he cared to much. There is no sign that he really has questioned this.
(Cohen): But as well-meaning as many of these attempts were on my part, some were certainly just too much…too outrageous and simply lacking in compassion and a deep appreciation of what is actually involved in change at the deepest level.
(Sand): Good, but I wonder if someone told him to say that.
(Cohen): More often than not what is needed is simply more love and encouragement, not more shocks, challenges and confrontations with one’s own division. There were times of course where strong challenges are called for and many former students have reminded me of many ways in which I did help them to reach breakthroughs through harsh tactics…but there is no doubt this happened too often, and more often than not it caused more harm than good.
(Sand): Good, I believe this.
(Cohen): I was a revolutionary, and publicly declared myself as such…ant that’s why many of you came to me. But that can no longer be an excuse for my own insensitivity and at time ruthless attempts to force deep changes to occur. Again I deeply apologize to any of you who suffered unnecessarily because of this.
(Sand): Good.
(Cohen): Over these two years away, I have come to appreciate with growing regret that the hierarchies that I had used as a teaching tool gradually over time became ossified and rigid, becoming for some not too different to being held in a straightjacket or a prison….Finally what has been hardest for me has been facing and coming to terms with the fact that I have let down so deeply and betrayed my former students whom I was closest to, those former senior students who had trusted me with their lives and souls and who gave so much to make it possible for the promise of Evolutionary Enlightenment to come alive in the world….I who had demanded so much was, when my turn came, seemingly unable or unwilling to do the very thing I had asked from them. I am so ashamed about this and my public apology was really meant for them.
(Sand): From this paragraph I am shocked to learn that he really does care about his most senior students and feels ashamed and sorry for his behavior which hurt THEM. But what about the hundreds and probably a couple of thousand other students who were never senior students. Some were peripherally involved and yet terribly traumatized. Some still are. The outcome of the children who were reared in the community is another story—not a pretty one.
(Cohen): Almost 2 years after my fall from grace and the collapse of EnlightenNext, I still care as much as I ever did about most of what I taught and a lot of what I stood for. I am committed to giving the rest of my life to trying to make good on it all.
(Sand): I hope that does not mean having another go at being a teacher.
(Cohen): What that will mean, of course, remains to be seen. Through this process of coming to terms with all that has happened, so many important questions have understandably arisen. As I make progress in my inquiry, I will be writing more about it here. I still love you very much and hope from the bottom of my heart that you will find it in yourselves to believe that even Gurus with big egos can find the courage and humility to change. I know in “Embracing Heaven and Earth,” I boldly stated that once Enlightenment has occurred, an individual gets frozen in their development—that from then on their evolution actually comes to a halt forever. I am committing the rest of my life to proving myself wrong.
(Sand): So did he miss the whole point which is for him to question whether he did in fact love us, or whether we were characters in the working out of his unconscious mind. There is no question that Andrew could now again engender love, trust and adoration from followers. My question is whether he actually could sit down with another person he had hurt and imagine their experience at times when he had hurt them. Whether his ability to intuit, includes the ability to attune to another’s mind and feel with them, in a way which has nothing to do with him.
I am thinking about my past experiences with Andrew. There were many times when meeting with Andrew, particularly in private, left me overwhelmed with feeling his love. There is nothing new in this. And I remember too that these experiences alternated with severe rejection. An ex-senior student once described to me a pattern of behavior in Andrew. One minute he might be meeting with apparent deep compassion with a student and then the next moment he would turn around even mid-sentence and attack them viciously. I have tried to understand this pattern. I long ago rejected the interpretation that this pattern represented his deep acceptance of my connection with God and rejection of my turning towards ego. Now I believe this patterns of behavior fits with a picture of narcissism.
The question I ask myself now is this. If he was/is narcissistic then what is going on when he appears to meet us in deep compassion and we feel accepted at the depths of our being like a true friend? One hypothesis is that when he is at his very sweetest he is not really there as a real partner with intact theory of mind, who really empathizes with us as another person on the other end. He is instead interacting with a source of possible fodder for his own affirmation. My hands are shaking when I write this because at an emotional level this goes against what I felt so deeply. But logic tells me that this hypothesis is worth pursuing.
The intimacy and connection are something we felt/feel because we have opened our hearts to him. He on the other hand is only able to turn around and attack us because he did not in fact feel that connection and heart opening with us in the first place. WE might perceive it as a switch in his affections, but for him there was not real affection involved in the first place. I am suggesting that on the occasions when he failed to get what he wanted out of a meeting with us then there would for him be no logical or emotional inconsistency with his attacking us shortly thereafter.
I am concerned for the large numbers of ex-students of Andrew, who still look toward him with devotion and may well ask him to return as their teacher. I leave room for the possibility for every person to grow and develop. I wish for Andrew to see himself more clearly and to become more connected to the people in his life. I want to acknowledge that for someone who thought he was perfect it is an extraordinary shift for Andrew to have questioned his motivation at all. I welcome this. Yet is is not clear whether this apology is the beginning of a deep-seated change within or whether it is an unconscious movement to repeat the pattern of seeking influence and control over others. Time will tell. The jury is still out.
As I assimilate what I have learned from reading Marlowe Sand’s book and digesting her responses to Andrew Cohen’s apology, I have gained more understanding about my initial determination that Cohen’s apology was neither truly sincere or manipulative. When I wrote my essay about this in May, 2015, I chose to classify his apology in the category “other” (cf. [1]) for lack of any better way to conceptualize where it belonged.
However, upon further reflection in consideration of all that Marlowe has conveyed, I must say that I agree with Frank Visser in his comparison of Cohen with Mark Gafni, as Frank has this to say about Cohen’s apology:
He showed some self-insight in that he had been “hiding in transcendence,” neglecting the dimension of “vulnerable humanity” both in himself and others. Though I have never been a fan of Cohen’s philosophy, I think this was a great gesture and a step in the right direction. [6].
Although I also think that Cohen had devastating negative effects on far more people than Gafni did, I am willing to give Cohen the benefit of the doubt for at least making what I now construe as a reasonably authentic apology.
Yes there could have been more substance to his apology, as Marlowe Sand describes, in particular having more insight and awareness about his condescension towards women, feeling and conveying sincere remorse for the multitude of people he had devastating effects upon—aside from his focus on his senior students, and conveying real life examples of his mis-deeds. However, Marlowe affirms a number of Cohen’s statements as being “real,” and she conveys to us in no uncertain terms that she appreciates the crux of his apology. If Marlowe Sand, after all she has suffered for 15 years being under the control of Andrew Cohen in practically every aspect of her life, can rise to the level of being able to appreciate at least “some” of Cohen’s apology, then certainly I can do the same.
I don’t think it is insignificant that Cohen has stayed out of the limelight for the past 2 years, and I would like to think that he is truly taking stock and doing his personal retreat work. Of course I could be wrong, and I think Marlowe’s concerns are legitimate that a large number of Cohen’s ex-students still look toward him with devotion and may ask him to return as their teacher. But all things considered I must agree with Marlowe Sand that “for someone who thought he was perfect it is an extraordinary shift for Andrew to have questioned his motivation at all.”
As Marlowe says, “Time will tell. The jury is still out.”
1) See Elliot Benjamin (2015). Andrew Cohen’s “Apology”: Truly Sincere, Manipulative, or “Other”? Retrieved from www.integralworld.net
2) See William Yenner (2009). American Guru: A Story of Love, Betrayal, and Healing: Former Students of Andrew Cohen Speak Out. Rhinebeck NY: Epigraph Books.
3) See Marlowe Sand (2015). Paradise and Promises: Chronicles of My Life with a Self-declared, Modern-day Buddha. O books. Winchester, UK.
4) See in particular Hal Blacker (2013). The “A” List of Andrew Cohen: A Catalog of Trauma and Abuse; Be Scofield (2013). Integral Abuse: Andrew Cohen and the Culture of Evolutionary Enlightenment; William Yenner (2011). Cut From the Same Cloth: Scientology and EnlightenNext. Retrieved from www.integralworld.net
5) Andrew Cohen’s 2013 apology (as well as his 2015 apology) is no longer available online at the site it originally appeared at. However, Marlowe Sand included in her book (cf. [3], p. 251) the following statement from his 2013 apology:
Some of my closest students have tried to make it apparent to me that in spite of the depth of my awakening, my ego is still alive and well…when I was being asked to face my own ego by those who were nearest and dearest to me, I resisted. And I often made their lives difficult as a result. I’m aware that many of my students over the years have also been affected by my lack of awareness of this part of myself. As time passes I intend to reach out and engage in a process of dialogue with those of you who would like to.
6) See Frank Visser (2015). What’s Love Got to Do with It: Love Guru Marc Gafni Under Attack after New York Times Publications. Retrieved from www.integralworld.net
]]>BENJAMIN, ELLIOT (2013). e Creative Artist: Mental Disturbance and Mental
Health . Raleigh, NC: Lulu Publishing Services. 452 pp. ISBN 978-1-48340-355-
7, Paperback, $24.50. Reviewed by Ruth Richards.
This unique book, e Creative Artist, Mental Disturbance and Mental Health,
by Elliot Benjamin, Ph.D., should perhaps be classified as autoethnography
and is a very readable gift of personal sharing and artistic promise. Here one
.....]]>BENJAMIN, ELLIOT (2013). e Creative Artist: Mental Disturbance and Mental
Health . Raleigh, NC: Lulu Publishing Services. 452 pp. ISBN 978-1-48340-355-
7, Paperback, $24.50. Reviewed by Ruth Richards.
This unique book, e Creative Artist, Mental Disturbance and Mental Health,
by Elliot Benjamin, Ph.D., should perhaps be classified as autoethnography
and is a very readable gift of personal sharing and artistic promise. Here one
finds theory, personal contemplation over a few decades, sections of a novel,
and vivid in the moment notes from his creative and adventurous son, also
leading the creative artistic life in a society that doesn’t always encourage it.
Both are clearly persons of exceptional creative talent and complexity—and of
persistence and strength. As Dr. Benjamin (with Ph.D.’s in both math and
psychology, who taught math for over 20 years at the college level) said years
before, in a letter to his yet unborn son, he wished him the ‘‘joys of being
‘yourself, i.e., to know what it means to life an authentic life where you are true
to your own dreams and ambitions’’ (p. 61).
As Dr. Benjamin, this creative artist/mathematician/musician/philosopher/
psychologist/author, also found, this authenticity can be vital to one’s
connection with others (in his own life and his son’s), and can lead to
exceptional moments whether with family, friends, or one’s life partner. Surely
there is health here. Yet making it a mainstay of one’s life, in our consensual
reality and our current culture, may not always be that easy. Out of this
conviction came Dr. Benjamin’s own skill learning and educational,
psychoeducational, and supportive program, the Natural Dimension; we see
alternative trajectories along a humanistic and transpersonal path. Among
other things, his overarching model became a viable aftercare path for some
patients leaving more traditional mental health programs.
Yet finding and sharing our self and path is hard enough for anyone whose felt
creative task is to reveal an inner world that even minimally challenges the
status quo; there are many forces arrayed against such a change in the larger
world, as this reviewer, for one, has expressed. Dr. Benjamin expresses this
powerfully in terms of his ‘‘reality argument,’’ where we are called to do the
practical, the lucrative, and very much the expected. He, himself, could teach
mathematics—dealing with the practical world—although, there too, he did
find colorful ways to inject creative methods into the programme. Not to
mention, in his larger life, philosophy and music. He found it quite natural to
inject another world, one of elegance and beauty, into the picture. He was
drawn to a wish to ‘‘taste of ‘higher’ worlds.’’ Yet the role, and the task, isn’t
always one of beauty, nor of escape. Nor is everyone at a higher level of
psycho-social-spiritual development.
Further, not every creator is called to take this particular path. Sometimes the
creative artist, committed to a path of vision, takes a role of truth teller more
than visionary, poignant teller of bitter truths that others prefer to skip, or
deny (along with most of society). He or she, as an artist friend of mine put it,
is often ‘‘the canary in the coal mine.’’ Not only more sensitive to what’s going
on, but also at times more readily injured by it as well Some real dangers and
challenges here. It surely can help to have support, and role models and
encouragement, as we indeed see here in what father is offering son. But one
must not think all is euphoria. There may be such moments, as the son
delightfully remembers from his youth. But it is not the whole picture.
All the more, then, strength and overriding purpose are needed if there are
mental health challenges to address, particularly in a world that can
pathologize difference, of whatever sort, healthy or not. Some of us have in
fact asked for a different definition of ‘‘normal,’’ including this author in the
steps of the legendary Frank Barron, since what is most common (one type of
norm) may not in fact be the healthiest — at all. This is very much what we are
hearing here, in terms of conformity to social norms. At one extreme we have
a mindless automaton, following conventional rules, by contrast with a very
present, aware, insightful, and risk-taking artist, a higher ideal if not a statistical
‘‘norm.’’ Dr. Benjamin has called society, at times, crazy. I have done so a
well!
Fortunately, the author does not fall into the trap of romanticizing illness (or
saying it is de facto creative in itself) or that all who create are ill or vice
versa. It is definitely not ‘‘the sicker the better.’’ He acknowledges there are
many roads to creativity. One must still keep in mind that there are many
types of creative artists and motives for creating. Of particular importance,
health is the overriding message. How tragic the consequences for some who
have idealized illness. Yet on the upward path, some may indeed reach
heights of Maslow’s (1968) ‘‘self-actualizing’’ creativity, as suggested here, it
is not all. Dr. Benjamin mentions that, at times, ‘‘a fine line’’ may divide the
creative artist and the mentally disturbed person. Yet, let us look more
carefully here.
Some truth can be found, but it seems vital to clarify that this is in comparing
specific pieces of a complex human picture, and not necessarily the totality, or
the delicate balance and conditions that produced them. Dr. Benjamin was
once fortunate to converse at great length with philosopher Ken Wilber (1995),
who was intrigued by Dr. Benjamin’s thinking and also influenced it
considerably, including the Natural Dimension. This contact led to many of
Dr. Benjamin’s later publications. However, Wilber has also made this same
point in a spiritual context: Advanced spiritual development may at times
resemble more primitive patterns (pre/trans fallacy), yet the context and
significance can be totally different. They may all look like oranges but in fact
they are apples and oranges.
Despite findings of above average rates of psychopathology (and particularly
mood disorders, with evidence as well for schizophrenia spectrum disorders)
among artists—often among the eminent (thus we cannot automatically say the
same for everyday people) —the literature does suggest high creativity is much
more about health than illness. (Plus there are other paths, not at all about
illness.) Here, creativity tends to peak—where mental disturbance is present a
all, in individuals or in families, especially in research on people in the bipola
and schizophrenia spectra. Creativity peaks during during better functioning
periods. And, as a corollary, getting treatment can—not only be lifesaving in
some cases but also can— enhance creativity
One may speak of creativity as a compensatory advantage, and there is good
everyday creativity data we (e.g., colleagues Dennis Kinney and others at
Harvard) and others have to support this. Indeed it can be a major strength in
situations of risk, allowing perspective, alternatives, transformations of
a situation; one finds a delicate balance of the forces that open one to
inspiration, balanced by the control factors or executive functions that can
shape this to adaptive ends. This is what the neo-Freudians earlier called
‘‘regression in the service of the ego.’’ This is the Freudian ‘‘ego’’ used in the
sense of ‘‘adaptation to reality.’’ Dr. Benjamin mentions ‘‘ego strength’’ (p. 57)
as well. This is the ego-strength we wish some teenagers would show more
strongly, or show more often. In high level creators, the overarching control
may fluctuate, allowing certain flights of fancy, and even, and by deliberate
choice, the taking of psychological residence (think of the novelist) in
alternative worlds. Yet ultimately, and when needed, that loose overriding
control is always there.
What we see in these rich personal journeys and reflections are, first, Dr.
Benjamin’s own thoughtful conclusions on relations between arts, menta
health, mental disturbance, spirituality and other factors. A positive
trajectory of overcoming is seen to include the author’s brother, who is
viewed as manifesting ‘‘self-actualization’’ despite obstacles. This includes the
interesting issue, later overcome in a more transpersonal context, of what he
calls egoism. Let us put our pointer again on ‘‘ego-strength.’’ It is not or need
not be ‘‘all about me,’’ but we do need to strengthen qualities of the self.
Beyond this, mental health is here defined, in higher developmental terms,
ethically and psychologically. For this purpose of intentions, we are not
necessarily speaking of all artists, either, but of a more prodigious subset—
and could it perhaps even be a trajectory for some artists who persist? Som
of us have thought so. See Maslow’s (1968) ‘‘deficiency’’ and ‘‘being’’ needs
as applied to creativity.
Dr. Benjamin’s essays of personal reflection elaborate on certain themes, and
very much his semi-autobiographical novel (excerpted here). How marvelous
too, to have a mathematician-author who is also an artist and a musician
and…can speak to us in so many languages and ways of knowing. Then, in son
Jeremy’s story, one hears in his own words (and lived in his own way), the
brave persistence in a tough and demanding first year as creator and actor in
Los Angeles, and the eccentricity that both helps him to cope and may be
misunderstood. Let one not automatically pathologize anxieties, late nights,
excess energy–but celebrate it, particularly when enacted in the service of those
overriding goals. Here, indeed, is that balance of creative talent with the
strength, grit, gumption, and resilience, to succeed. And, happily, some initial
success, after the first year. May this prodigious young man go on to do very
well indeed.
Ultimately, Dr. Benjamin wisely recommends education, counseling, and
supportive and insightful resources (such as he himself started) to provide
skills, perspective, encouragement and understanding of a different artistic
world and worldview, perspectives which a creative artist may need and no
find as readily in the conventional world – a world which must still be
maneuvered successfully. For Dr. Benjamin, as well, being a ‘‘philosopher in
life’’ has helped him see a larger picture and keep his own perspective. He
hopes that this heartfelt, authentic, and very personal series of accounts,
contextualized with his own exploration of the field, its literature, and its
voices, has also helped in ‘‘loosening the bonds between the creative artist and
mental disturbance, and of honoring my family heritage’’ (p. 399). I would say
it has surely done so. May its insights further help open the doors to creativity
for all of us.
REFERENCES
BARRON, F. X. (1968). Creativity and personal freedom . New York, NY: Van Nostrand.
DENZIN, N. K. (2014). Interpretive autoethnography . ousand Oaks, CA: Sage
Publications.
GOODWIN, F., & J AMISON, K. R. (2007). Manic-depressive illness: Bipolar disorders and
recurrent depression . New York, NY: Oxford University Press.
MASLOW, A. (1968). Toward a psychology of being . New York, NY: Van Nostrand.
RICHARDS, R. (Ed.). (2007). Everyday creativity and new views of human nature .
Washington, D.C.: American Psychological Association.
RUNCO, M., & RICHARDS, R. (Ed.). (1997). Eminent creativity, everyday creativity, and
health . Stamford, CT: Ablex.
SASS, L., & SCHULDBERG, D. (Guests Eds.). (2000–2001). Special issue: Creativity and the
schizophrenia spectrum. Creativity Research Journal , 13(1) .
WILBER, K. (1995). Sex, ecology, and spirituality: e spirit of evolution . Boston, MA:
Shambhala.
e Author
Elliot Benjamin , Ph.D., is a philosopher, mathematician, psychologist,
musician, writer, teacher and counselor, with Ph.D.’s in both psychology
and mathematics, with numerous publications in humanistic and transpersonal
psychology, and over 20 years of college teaching experience in mathematics.
He is currently Director of online education in transpersonal psychology at
Akamai University.
e Reviewer
Ruth Richards , M.D., Ph.D., is a professor at the School of Psychology,
Interdisciplinary Inquiry Specializations of Consciousness and Spirituality and
Creativity Studies at Saybrook University in Oakland, CA. She is the editor of
Everyday Creativity and New Views of Human Nature , and the co-editor of
Eminent Creativity, Everyday Creativity, and Health .
* * * * * * *
Journal of Transpersonal Psychology, 2015, Vol. 47, No. 1
]]>
Divine Light Mission: Cultic Commitment Over A Lifetime Story
Elliot Benjamin, Ph.D.
Accepted 30 August 2015
ABSTRACT
This article describes the commitment to the religious organization Elan Vital (whose original name was Divine Light Mission) over a lifetime
through the author’s intermittent correspondence with a childhood friend for over 40 years. Elan Vital was rated by the author in.....]]>
Divine Light Mission: Cultic Commitment Over A Lifetime Story
Elliot Benjamin, Ph.D.
Accepted 30 August 2015
ABSTRACT
This article describes the commitment to the religious organization Elan Vital (whose original name was Divine Light Mission) over a lifetime
through the author’s intermittent correspondence with a childhood friend for over 40 years. Elan Vital was rated by the author in his previous
work has having a moderate degree of cult dangers. This article gives a firsthand account of what it is like to be a lifelong committed member
of this organization, from the author’s perspective.
Keywords:
Elan Vital/Divine Light Mission, Guru Maharajji/Prem Pal Rawat, Premi, cult dangers .
Corresponding Author:
Elliot Benjamin, Ph.D.
Email: ben496@prexar.com
Pinnacle Sociology & Anthropology
(ISSN: 2360-959X)
http:/www.pjpub.org
© Author(s) 2015. CC Attribution 3.0 License.
Research Article
Introduction
I first learned about Divine Light Mission in 1973 when I was
23, from my boyhood friend Richie. Richie and his fiancé Linda
were thoroughly enamored with the guru of Divine Light
Mission, 14-year-old Guru Maharaji¹ from India, who was
proclaimed by Divine Light Mission as being the lord of the
universe. Over the next 5 or 6 years, I engaged in a number
of get-together’s with Richie and Linda as they would
proselytize about their guru to me, frequently with my ex-wife
Diane, as well as letters and phone conversations with Richie.
Richie and I would become immersed in our heated, extensive,
and repetitive guru-no guru dialogues and arguments. Of
course neither one of us would convince the other to think
about the legitimacy of Maharaji’s guruship any differently,
but this never discouraged Richie from always trying his very
best to “show me the light.”
These guru-no guru conversations came to a natural hiatus
soon after my son Jeremy was born in 1981. Richie and Linda
visited us when Jeremy was a baby, continuing to proselytize
to us about their guru, and I soon became quite taken up with
being a new father and earning my living as a mathematics
instructor and soon mathematics professor. I had no further
contact with Richie for the next 9 years, but in 1991 I visited
him and Linda in Montreal with my 10-year-old son Jeremy.
Richie and I got back to our old guru-no guru dialogue during
this visit, as it there were no 9 year break in the action. But
after one or two letters between us over the next year or two,
our communications came entirely to an end.
That is, they came to an end until I received a phone call from
Richie 17 years later in 2008 (see my essay
Guru Maharaji Gets
Sat-Sang
below). And then another 7 years passed without
any further contact with Richie, until a few weeks ago my now
33-year-old son Jeremy, who I was visiting in Hollywood,
California, Googled Richie and Linda for me and found out that
they were currently living close to Los Angeles. One thing led
to another, I had a brief conversation with Richie, and I agreed
to call him soon for us to “catch up.” I am expecting that “soon”
will be tonight, and I am bracing myself for another guru-no
guru long conversation with Richie, over 7 years after our last
one. Of course I have no doubt that this will accompany our
usual topics of conversation: the “love and magic”² of Richie
and Linda’s relationship, Richie’s advice to me about my own
romantic relationships, our debates about the reality of life
after death, my philosophy of natural dimension³, etc.
In my
Modern Religions book (Benjamin, 2013), I concluded
from my tri-perspective experiential analysis of Divine Light
Mission, based upon what I learned from my friendship with
Richie and Linda, that Divine Light Mission has a “moderate”
level of cult dangers⁴. One of the key ingredients of my
analysis was Guru Maharaji’s personal charisma to induce
people to accept a 14-year-old overweight Indian kid as the
lord and master of the universe. I rated Divine Light Mission
and Guru Maharaji extremely high in the cult danger scale that
I utilized in the categories of wisdom claimed, wisdom
credited, and dogma⁴. I also placed Divine Light Mission in the
pre-rational level of Ken Wilber’s (1995) consciousness
continuum, and rated him high in the category of wealth⁴. The
way that Guru Maharaji accumulated his wealth through the
donations of his devotees while so many of his devotees lived
lives of poverty is a major source of ethical concern about
Divine Light Mission/Elan Vital, as described on a number of
websites⁵.
Thus this reunion conversation that I will be having with
Richie is especially interesting to me, given the conclusions I
have previously come to about the cult dangers of Divine Light
Mission/Elan Vital. However, before I engage in and describe
this reunion conversation with Richie that I will soon be
having, I will convey what I have learned about Guru Maharaji¹
and Divine Light Mission through my intermittent but lifelong
friendship with Richie, from the following essays I have
written in the 1970s and 1980s.
On Guru Marajji (Divine Light Mission)
(1977)
I have been somewhat familiar with Guru Maharaji and Divine
Light Mission for about 6 years now. It all began when my
boyhood friend Richie and his wife Linda became devotees of
Guru Maharaji, back in the summer of 1971. At that time
Maharaji was 14 years old and was proclaimed throughout
India as a living perfect master, and moreover-one who was
destined to bring peace to the world, in his own lifetime.
Thus Maharaji came to America, and brought forth a new
religion to many confused and disenchanted young Americans.
At the time, Maharaji was quite a comical figure to our
population at large, as to me he looked like nothing more than
a plump, egocentric fat boy who had barely started to shave.
This was the lord and master of the human race? A direct
incarnation of God? The only thing that kept me from totally
discounting this new religious fad was the respect I have
always had for my friend Richie, whom I have known since i
was 11 years old. I was sure that Guru Maharaji was just
another phase that Richie was going through, and would soon
pass over. Well this phase has lasted for 6 years so far, and if
anything it appears to be stronger than it ever was. As I have
Pinnacle Sociology & Anthropology ISSN: 2360-959X Page 2
kept up my friendship with Richie, I have had no choice but
to take Guru Maharaji more and more seriously, until I now
feel that he deserves a proper place in my book on modern
religions, right after Scientology, EST, and The Unification
Church⁶. Maharaji is now a young man of 19, a husband and
a father, and an outcast from his family because of his
marriage to an American girl. Guru Maharaji is still somewhat
on the plump side, but Divine Light mission has thousands of
ardent followers all over the world, and is growing stronger
every day.
A devotee of Guru Maharaji is called a “Premi,” and to a
Premi-Guru Maharaji is everything that Christ was to his
followers, two thousand years ago. Premis proclaim that Guru
Maharaji is in a class with Buddha, Moses, and Christ, and that
Maharaji is here to complete the job that these other perfect
masters before him began. This claim is very similar to that
made by the followers of Reverend Moon, founder of The
Unification Church, except for the fact that the Unification
Church does not always specifically name who this living
perfect master is⁷. I have been to an ashram (the spiritual
place of worship for Premis) twice, and attended Millenium,
the vastly publicized Guru Maharaji festival in Houston’s
astrodome in November, 1973, while I was living in Houston,
Texas. So what have I to say about all this? Well, to me-Guru
Maharaji is still little more than a plump, egocentric, fat boy,
but one who is growing up fast. I have no way of knowing
whether Maharaji is truly a living perfect master, or if a perfect
master has ever existed. But for what my opinion is worth, I
do not believe that Maharaji is any more or less human than
you or I or anyone else in the world. Maharaji is from India, a
country that believes wholeheartedly in reincarnation, gurus,
and transpersonal beings. I think that Maharaji himself
believes that he is all that he is proclaimed to be, for he has
heard nothing else for his entire life here on Earth.
At first glance it might seem totally remarkable that hundreds
of young men and women in a New York ashram can bow
down on the floor to a picture of Guru Maharaji and
wholeheartedly proclaim that he is the Lord of the Universe.
But if you stop and think about it, is this really any more or
less silly than the advent of Christ-or of any other religious
figure who is said to be something more than a mere human
being? What is important is that all these people really do
believe that Maharaji is Lord of the Universe, and they find
peace, comfort, and even salvation in this belief. They have
given their lives to Guru Maharaji, just as their ancestors have
given their lives to Christ. It is tough to live a life without some
sort of supernatural religious belief, and it gets tougher as one
gets older⁸. I myself have changed from a confirmed atheist
to a confused agnostic. What is attractive about Maharaji is
exactly what makes him appear so unlikely to be a true perfect
master-his age, his fatness, and his richness. His followers defy
all logic and all intellectual use of the mind, and what better
way is there to stress the unimportance of the mind than to
accept that which appears wholly non-rational, unreasonable,
and contrary to everything the mind has always thought?
Accepting Guru Maharaji is wholly an act of faith; there is
nothing logical about it, nothing whatsoever;. There is Sat-
Sang-a nightly get-together where Premis take turns in
opening up their hearts about all that Guru Maharaji has done
for them in their lives. There is Meditation-a special kind of
meditation which is taught by the “Initiators” in a 7 hour
experience called “Receiving Knowledge,” whereby one is fully
initiated into Divine Light Mission and promises to accept
Guru Maharaji as one’s savior for the duration of the universe.
And there is Service-the job of every Premi in his/her
continued efforts to spread the presence of Maharaji to others
and to do his/her share in bringing peace to the world. The
message of Guru Maharaji is pretty straightforward Eastern
Buddhism; one must get rid of one’s individual personality
and join with the universal and everlasting. What is different
about Guru Maharaji is the physical presence of Maharaji
himself, which supposedly makes it actually possible to
achieve or at least move close to this state.
I have grown, over the years, somewhat fond and tolerant of
Guru Maharaji. I like the Premis I have met, and I definitely
feel a lot of love, warmth, and good will in the ashrams that I
have attended. I suspect that this religious ardor and warmth
is very much lacking in many of our more traditional places
of worship that exist in Judaism and Christianity. I believe that
Maharaji will appear as a much more legitimate figure to
prospective followers when he is in his 20’s compared to
when he was 15, and there is really no telling how far Divine
Light Mission will spread. Followers of Guru Maharaji, aside
from the inner peace of mind they gain, become part of an
extended family that gets together every evening for Sat-Sang.
There is much joy and happiness in an ashram; peace and love
certainly do reign throughout. It seems to be a very human
quality to need a transpersonal human figure to put one’s faith
in, and perhaps we are today witnessing a rebirth of authentic
religious feeling in this country.
But for the record-in my opinion Guru Maharaji is a pleasant
trap. A trap to give up on life-to give up prematurely. There is
no more self-imposed growth once one accepts Guru Maharaji,
for there is no more self. It is tempting, but luckily not
tempting enough for me. I believe there is a better way. A way
in which all of our potential mental abilities can be respected
and utilized, but a way in which our spiritual centers do
transcend our minds. This way rejects any one human figure
as being the representation of God. This way stresses that the
path is where it always was-within the individual, and then
across to other individuals. This way explores anything and
everything in life that is worth exploring, and calls it
knowledge and experience. This way in not new- it is as old
as Socrates and it persists in some of us die-hards, some of us
who love life too much to ever give in to the conventions of
repressive society or to the dogmas and mind-destroyers of
proselytizing religions, whether they be modern or traditional.
I am by no means saying anything unique when I talk about
this way, for I am merely following in the line of a rich cultural
heritage, but I do wish to stress that my life will be lived to
indeed do my share to see that this line does continue. Natural
Dimension has no need of a name when one is in the woods,
doing theoretical mathematics, playing the guitar, or watching
a bird. However, Natural Dimension does have need of a name
when one is sitting in a mind-molding classroom, working at
a mind-lulling job, and witnessing a mind-shattering religion.
And so I have decided that more-much more-of my philosophy
of Natural Dimension will be making its way from my own
personal self to the selves of others-as an alternative, an
alternative to everything “un-natural” that is around you³.
Pinnacle Sociology & Anthropology ISSN: 2360-959X Page 3
Letter from Richie
(1978)
What more can I say about Guru Maharaji? I didn’t intend to
write a second essay on Guru Maharaji but I have just received
a letter from my aforementioned friend Richie, in which there
contains such a beautiful illustration of the experience of being
a Premi, that I wish to share it with everybody who is
interested in learning more about Guru Maharaji and Divine
Light Mission. I know that Richie will not be offended by my
inclusion of parts of his personal letter to me in my book, and
to give any sort of an authentic account of what it means to
be a devotee of Guru Maharaji the message must come straight
out of the horse’s mouth, so here it is.
“Dear Elliot,
Well it seems to be about that time again; it’s never an
easy thing to write to you because I can’t just scribble
off a “How are you, I am fine” letter. But since I’ve
started I’m sure the momentum will carry this
through….Linda and I just got back 3 days ago from
Rome, Italy. We had a 5 day festival there with Guru
Maharaji. 15,000 people from 50 different countries,
and simultaneous translating headphones in 15
different languages. So much love-such concentration
on one person. 15,000 hearts meeting just to see him
smile. Love is all there is and love is infinite. God is love.
I’m learning that in my relationship with Linda I only
glimpse the possibilities of love; but with Guru
Maharaji I can’t even comprehend the magnitude. I see
him totally transforming so many souls that for me to
not call him Lord or Master would be a gross
ingratitude. And I warn you, Elliot, if you dare write
anything about Guru Maharaji without first receiving
knowledge, all you’ll be doing is drawing a paint by
numbers picture without the colors. It’s like looking
into the wrong side of binoculars; all you’ll be defining
is something that’s too far away for you to grasp. The
further I dive into Maharaji the less I understand and
the more I realize that the key to God Realization is to
experience it and not to intellectualize about it. You can
read all the books you want about love but that will
never give you the feeling you get when you look into
Diane’s eyes. And the feeling I have with Linda is
nothing compared to the love Maharaji has in store for
me. I’m learning that the more i love him and open
myself up to him-the more I feel that love inside of me.
The more love I “feel” the more love I can “share”….I
haven’t written a song for months; in fact, this is the
first time I’ve taken pen in hand for this length of time.
But I feel totally satisfied-Maharajji must be filling me
up-he “is” filling me up….love to Diane-write.
Tell me everything soon,
Richie”
Well-that’s Richie. This is the kind of thing I’ve been hearing
from Richie for the past 6 years. In regard to the possibility
of me ever “Receiving Knowledge,” the last time I saw Richieabout
7 months ago in New York, I told him that I would be
curious to experience Receiving Knowledge. His response was
that before I would be able to Receive Knowledge I would have
to be ready to accept Guru Maharaji as Lord of the Universe.
In other words, Receiving Knowledge only works if you
already have utmost faith in Guru Maharaji. Thus, I cannot see
how I will ever be able to experience Receiving Knowledge.
But it is true what Richie says about my not Receiving
Knowledge being a severe detriment for me to convey an
authentic picture of Guru Maharaji and Divine Light Mission.
This is much of the reason why I am including this excerpt
from Richie’s letter and writing this follow-up essay to
On
Guru Maharaji
. I hope that Richie has added significantly to
your understanding of what it is like to be a devotee of Guru
Maharaji.
NOTE:
For a fictional portrayal of the experience of becoming
a devotee of Guru Maharajji, see
Excerpts from “The
Maturation of Walter Goldman,”
particularly the excerpt
entitled
Alienation, in my Modern Religions book (Benjamin,
2013), where the main character Walter is modeled after
myself, and Walter’s friend Zachary is modeled after my friend
Richie.
Diane Gets Sat-Sang
(September 22, 1978)
Well I have finally seen Richie again, as myself, Diane, Richie,
and his wife Linda got together for an interesting evening in
New York a few nights ago. Guru Maharaji is now going on 21
and has two kids. Richie and Linda were wearing Guru
Maharaji teashirts when we greeted them, with pictures of
Guru Maharaji all over their bedroom and living room. If
anything, their devotion seems to he growing stronger over
time. They want to follow Guru Maharaji all over Europe and
then follow him around the U.S.on his periodic Divine Light
Mission festivals. They no longer talk about making it in music
and they no longer talk about having kids and a family. Linda
seems so spaced out on Guru Maharaji that she is not able to
be interested in or relate to anything that Diane and I say
about our own lives. She indeed seems to be falling into the
trap that I wrote about in
Moonies and Premies⁹, i.e. not being
able to see anyone else’s path as valid.
Of course this might be an inaccurate projection I am making,
but this was my experience. Richie was made Food Service
Coordinator for Divine Light Mission in New York City, and he
considers this to be a divine privilege and honor. However, I
am happy to say that my friendship with Richie is still intact,
as he was able to be quite open to both myself and Diane.
But what was most interesting for me during this evening was
to see Diane’s experience of receiving Sat-Sang from Richie.
Diane listened very attentively to everything Richie said. In
fact, she listened so attentively that Richie ended up saying
how beautiful it was to finally be able to relate his blissful
experiences of Guru Maharaji to a non-Premi who truly gave
him space to share himself, and that he was now willing to
give us credit for having something special and maybe he
could learn something from us. Believe me, coming from my
old friend Richie this was truly an incredible statement to
hear. For so many years, I have been trying to convey to Richie
that I too know something, and he finally has given me the
space to live and breathe on my own terms.
Richie realizes that the Premies in Divine Light Mission have
much to learn about love, as their own personal relationships
leave much to be desired, from Richie’s experience. Richie says
that Premies are only human, and he has many objections to
much that goes on in the Premi community. Richie feels that
Pinnacle Sociology & Anthropology ISSN: 2360-959X Page 4
his relationship with Linda is his way of learning how to love
Guru Maharaji, and he believes that marriage should be a
sacred path for all people to learn how to love. He says that
marriage gives you a direct mirror into your own self.
Yes-much of what Richie says is very true to me, as I have
expressed so many similar ideas in my own philosophy. But
then Richie goes on and talks about the necessity of having a
Master in order to find out who you are in your deepest self.
The Master knows where you want to go and sets up the
conditions for you to go in exactly that direction. Diane retorts
that she has no need or desire to have a Master, as she and I
learn from each other. Finally, Richie accepts that perhaps it
is our fate in life to walk a spiritual path without a Master, and
a dangerous impasse gets resolved. Diane says she is able to
understand everything that Richie has been saying, in a way
that not many people would be able to understand. Richie says
that he knows this and appreciates this greatly.
What did “I” do this whole evening? Well, I played the piano
a little bit (my old piano which I had given to Richie and Linda
as a wedding present), and I listened a whole lot. Richie and
Linda were totally flabbergasted at my silence after their
opening of their hearts about Guru Maharaji. They expected
me to argue and debate like I always used to. No-I no longer
have anything to argue with them about. That’s them- and this
is me. If we can all accept each other, there does not have to
be any one right or wrong way for everybody. Only time will
tell what is really “real,” and I think a few more years will be
necessary before we are able to formulate any definite
conclusions about Richie and Linda and Guru Maharaji.
Elliot Gets Sat-Sang
(March 17, 1979)
About 5 weeks ago I had a telephone conversation with Richie.
I must tell you a little bit about my circumstances in order for
you to fully appreciate the impact that this conversation had
upon me. Diane and I had just celebrated the 9th anniversary
of our falling in love, and as an anniversary present I gave
Diane enough money to go for a long weekend personal
retreat to Mendocino, California (we were living in the
Berkeley, California area at the time). This was a very
significant few days for us, and marked the beginning of a
whole new era in our relationship. However, it was a long few
days for me, and it felt pretty weird being in our five room
house all by myself. I went through many deep insights about
my life and our relationship, and soon before Diane was
expected home I decided to call my friend Richie in New York.
To really appreciate the meaning of my calling Richie, you
should realize that I have a general aversion to the telephone
and that I virtually always write letters rather than call
friends. Well, the phone call has cost me $20, so I figure I at
least better try to get another essay on Guru Maharaji out of
it, so here goes.
I didn’t tell Richie the details of what was going on between
Diane and me, but I did tell him that we were still in process,
and have not attained nirvana yet. This was all Richie needed.
For the next half hour I got Sat-Sang. Of course Richie and
Linda were the same, were living in total bliss and harmony,
were like little children with each other, etc., etc. We went
through our old guru-no guru controversy, and I totally made
up for having played the piano during our last Richie & Linda
get-together. I once again succumbed to Richie’s exhortations
and proselytizing excitement. It made me realize how long it
had been since I had actively searched for God. I told him that
I still believed in Natural Dimension; i.e. in Mutual-
Internalness¹⁰ between me and Diane. I told him that Diane
and I were almost “there.” Richie has been in on our journey
from nearly the beginning, and he understandably took my
reassurance with somewhat of a grain of salt. He stressed the
importance of sharing your self with a community of other
people, and I stressed the importance of making Diane into a
happy human being. I told him that this was my life work, and
if I could achieve this aim then I would also become a happy
human being. I asked him if he could accept that he had his
way and I had mine, and perhaps they were both O.K. His reply
was remarkably brilliant. He said that when I tell him that I
am living in love and bliss, then he would affirm my existence.
I had nothing left to say. I could not say that I was living in
love and bliss, and I could not say that I did not want to be
living in love and bliss. I said that it was “Internal”¹⁰ for me
and Diane to have children soon, and he minimized the
importance of this to me, saying that nothing in external
circumstances was necessary to be happy. I disagreed with
him in this respect, and we said good-bye in a very distant
place.
So what have I learned from my spiritual telephone
conversation with my friend Richie? I learned that I am still
vulnerable to authentic spirituality. It is still something that
is very much a part of me. I am still on a search, although I
spend much less time actively searching than I used to. But
it just takes a few real words from a real friend to remind me
of who I am. I need to conquer materialism first; this is the
only reason I have taken a rest in my search for truth and for
God. But I foresee that I will be moving again shortly, as the
quest for authentic spirituality is as valid in my life now as it
ever was. Thank you Richie for giving me Sat-Sang.
Richie Gets Sat-Sang
(April 15, 1979)
I’m now at a small private public beach in the town of
Mendocino, California, which I have just made into a nudist
beach. Natural Dimension is alive and well, as Diane and I
are recapturing who we truly are; God is Mutual-
Internalness¹⁰ Do you hear that‐Richie? Last weekend I wrote
you a letter which I don’t expect you to answer, for you are
too wrapped up in your Guru Maharaji to really see me for
who I am. But in my letter I said that although I love you both,
you and Linda are two of the most closed-minded people I
know. I said some stuff about truly being open and susceptible
to what life has in store. This is Natural Dimension, Richie.
Sometimes I’m uptight, and sometimes I’m uninhibited. I
seem to work in extremes. But put me in nature with Diane
for a few days, and I inevitably find out who I am. If I believed
in past lives, i would say I’m Albert Einstein reincarnated. Am
I happy? I’m on the road to being happy. I’m almost content,
and that is a nice place for me to be in. And whatever I have
achieved and will achieve in life, I have done it directly to life
itself. You my friend have cheated; although you think you
know all the answers, you know absolutely nothing. For I
know that I know nothing, and therefore I know far more than
you. You’ll say you know nothing, nothing except that Guru
Maharaji is Lord of the Universe. What kind of weakness is it
that needs a guru-the way you and Linda need Guru Maharaji?
Why is true love not enough for a human being? It is enough
for me-more than enough, and I am thankful to be alive. The
challenge, the mystery, and the ecstasy are all experiences in
life that I feel privileged to have had. I could not have
experienced one without the other. So now my friend-I am
giving “you” Sat-Sang. I don’t expect you to ever listen, and
that is why this is going down as an essay for my book-instead
of as a letter to you. I know you don’t care, but I am sorry that
I feel you aren’t there anymore. I’m no guru; I’m only a human
being. But it’s good to be alive.
Guru Maharaji Gets Sat-Sang
(May 31, 1982)
I truly do believe that this will be my last essay on Guru
Maharaji. I had lost touch with Richie for over 3 years, when
one night I came home from my Men’s group meeting and who
do you think was on the phone with Diane? It turned out that
Richie was still very interested in us and was not able to locate
us the past few years, but finally managed to find us through
a mutual friend. He had heard that we had a baby, and he
seemed to be genuinely interested in us all getting reacquainted.
And so we all got together in New York for an
evening last Thanksgiving, and then Richie and Linda visited
us at our home in Easthampton, Massachusetts for 3 days-just
a few days ago.
Well I would have to say that Richie and Linda will always be
our friends, but with serous limitations. Richie and I go back
a long time, and neither of us wants to throw that away, but
there is virtually no change in their effervescence over Guru
Maharaji. However, to do justice to their faith, I must say that
there have been some interesting and positive developments
in the Premi community. It turns out that Divine Light Mission
no longer exists as the organizational structure of the religion,
and that the term “Guru” has been dropped; he’s now referred
to simply as Maharaji. It seems that Maharaji honestly does
not want any more of the circus antics and fanfare that
overrided all the festivals in their old days. The whole thing
seems to be simplified to such an extent that it is nearly
impossible for anyone to locate Premies on their own, without
knowing someone first. I welcome these changes very much,
and I am nearly tempted to stop calling Maharaji a cult.
However, when I look at the intrinsics of what is really going
on, I nearly want to vomit.
To backtrack a little, when we got together with Richie and
Linda in New York last Thanksgiving, I was very affected by
the beauty and happiness that Richie and Linda had preserved
in their 10 year marriage. Diane and I have been through so
many struggles, and we have preserved the special ingredient
of our love, but we by no means live day-to-day life in the
honeymoon stage, which is how Richie and Linda seem to live
every day. I’ve had a long-standing jealousy of their
relationship, but it always was that Diane and I were on the
path of reaching our potential. Well this time it didn’t seem
like we were on too much of a path. I was having tremendous
difficulties in supporting my new family, my in-law problems
were intense, and we were not at all happy. Richie and Linda
were truly caring and helpful. I will always appreciate their
sincere interest and wisdom in regard to what I needed to do
to get back on the track. But the personal details are not
relevant right now; the point is that I felt a very real bond
with Richie and Linda; a couple bond. This couple bond was
the motivation that led me to acquire more learning
experiences with Maharaji. Richie had changed his tune about
Receiving Knowledge over Thanksgiving. He said that the
Initiators were giving out Knowledge much more easily now,
and that you did not first have to believe that Maharaji was
Lord of the Universe in order to Receive Knowledge. Soon
after our New York visit I wrote to Richie that I would like to
experience Receiving Knowledge-this summer in Boston.
After a few tumultuous months of bad communication about
this matter, Richie and Linda came to visit us, and Richie and
I talked and talked and talked. It was very draining on me-both
mentally and physically, and I knew that if I was going to keep
some kind of a friendship with them, I had to put a stop to
Richie’s unceasing unwanted advice that he would
continuously give me. And so it was left that I was going to
check out Maharaji for myself.
And now I have checked it out enough to satisfy any lingering
curiosity that can possibly still be in me. Receiving Knowledge
is just as it always was-you have to first open your heart to
Maharaji, as a child to the Lord. This is the only way in which
the meditation techniques of Receiving Knowledge have the
effect they are noted for¹¹ I verified this quite personally,
having spent a few intensive hours with an Initiator in one-toone
dialogue, both in the Hartford, Connecticut ashram and
in the car ride from Hartford to New York, and there is no way
that I could ever Receive Knowledge. It is so very obvious to
me how Maharaji fills the hearts of those who are at their
dead-end; those who do not have the courage to face life on
their own. I had a very sad firsthand experience of this while
spending the night in the apartment of a very pitiful Premi.
Enough! I can stand it no longer. I saw continuous pictures of
Maharaji in the slide show at the Hartford ashram-a fat
egocentric young man. I feel a repulsion towards him-and I
always will. Yes his followers are sincere, including Richie and
Linda. They believe that they have found God, and there is no
changing that.
But for me, I choose to meditate on my own special form of
God-the gift of my family, i.e. me, Diane, and Jeremy. I have my
own mission in life, and it is to be worthy of the gift I have
received, and to make all the potential beauty inside my gift
come out. To hell with everything else. I am now content, and
I foresee the day when I will be happy. If God is not in me, then
there is no God. For I have preserved the ultimate meaning of
religion: LOVE¹²
Afterword #1:
9 years later I visited Richie and Linda in
Montreal, Canada with my then 10-year-old son Jeremy in
1991, and it was evident that Guru Maharaji continued to be
the ultimate source of spiritual meaning and happiness in both
of their lives, as they were still living in total bliss and
harmony. As of 2005, Divine Light Mission is alive and well.
Afterword #2:
17 years after my 1991 get-together with him,
I received a surprise phone call from Richie in April, 2008,
after not having heard from him in all these years.
Within a few minutes it was the same old story-Richie urging
me to check out Maharaji and Receive Knowledge that is now
available through the guru’s CDs that can be done at home,
etc. Maharaji and Divine Light Mission both have new names
(check out Prem Pal Rawat and Elan Vital at
www.elanvital.org
) but the more things change the more they
Pinnacle Sociology & Anthropology ISSN: 2360-959X Page 5
seem to stay the same. As of May, 2011 I have had no further
contact with Richie or Divine Light Mission/Elan Vital.
Conversation with Richie 7 Years Later
And 7 years later Richie and I have had another conversation.
Yes Richie and Linda are still devotees of Guru
Maharajji/Prem Pal Rawat and Divine Light Mission/ Elan
Vital, but I must admit that I was very surprised (and relieved)
that this time Richie did not plagiarize to me about his religion.
We talked for an hour and fifteen minutes, and nearly all of
our conversation was about Richie’s music and art, his living
alone in California for the past 8 years while remaining close
with Linda, his very involving platonic friendship with a
woman who lives near him, his health challenges, and a bit
about my own artistic development over the past 20 years.
However, in passing Richie mentioned something about his
old friend, who I knew from Brooklyn when I was a kid, having
“received knowledge.” But it actually took a few prompting’s
before I got Richie to tell me about his current involvement
in Divine Light Mission/Elan Vital, as he was obviously much
more interested in telling me about the immense changes that
have taken place in his life. And when he did finally describe
to me what his current meditation practice is like, he
emphasized that the focus of his 2 hour daily mediation was
on going deep inside himself, without his usual plagiarizing
way of describing to me how sublime and magical Guru
Maharaji/Prem Pal Rawat was. What he actually said was that
Prem Pal Rawat now conveyed that his followers should not
focus upon him as an “inspirational speaker,” but instead
should go deep inside themselves to attain spiritual wisdom⁵.
When Richie talked about the meditative quality of his music
and art, I told him that this was similar to how I practiced
meditation myself with my mathematics and music, and that
this sounded similar to me to the practice of mindfulness
mediation¹⁶, which was what my fiancé Dorothy’s meditative
practice is. Richie listened to me well and did not minimize
what I was saying in the way that he always used to, as in the
past he would always try to convey to me how much “higher”
his meditation on Guru Maharaji was after “receiving
knowledge.” No-I must admit that there was no more
proselytizing in this conversation. All things considered, this
conversation felt pretty much like an old friend telling me
about his life, and included a description of his
religious/spiritual practice, but without trying to convert me.
I don’t know if Richie has truly become as mellow as he
sounded to me on the phone, in regard to accepting that others
have their own legitimate spiritual paths that are different
from his own. Perhaps in our next conversation Richie will
once again get back to his proselytizing/trying to convert me
mode of conversation. But for now, I must say that I was quite
surprised and relieved to feel that I reconnected with my old
boyhood friend Richie without our usual intensive guru-no
guru conversation. If anything, it makes me feel more open
to reevaluating Divine Light Mission/Elan Vital and Guru
Maharajji/Prem Pal Rawat. But to consider all possibilities,
perhaps this is what Richie wisely decided to do; i.e. to not
put pressure on me to “receive knowledge,” but rather to let
me gradually feel more favorable to his guru. However, at any
rate, I must give credit where credit is due, as at this time I
must honestly say that I feel relatively calm when I think about
Divine Light Mission/Elan Vital and Guru Maharajji/ Prem Pal
Rawat. Perhaps it’s time for me to check them out on the
internet and see what I can find out from what others have
experienced.
Afterword: 8/9/15
Before I check them out on the internet, I would like to briefly
describe what has taken place for me with Richie over the last
few weeks since out recent phone conversation. I received an
e-mail from Richie about a week after our conversation, saying
that he read my
Experiential Skepticism: An Exploration of
Mediumship and Life after Death
essay¹³, and describing his
and Linda’s experiences with “psychic surgery,” which I had
described from an experiential skepticism perspective in my
above article. In this e-mail, Richie said that he was focused
upon connecting with spirits in “this” lifetime rather than with
ones who have passed on, and he sent me some of his music
CD’s. We exchanged a number of e-mails over the next few
days, about topics such as him having a bicycle accident soon
after his recent move to Florida, and my remarking that I liked
his music and artwork that I had recently checked out. Richie
did mention briefly about how wonderful it was for him to be
able to connect with a spiritual master in his daily meditation,
but once again for the most part our communications were
not focused upon his guru.
However, in the last e-mail I received from Richie, which was
over a month ago, he asked me if I would ever have an interest
in coloring in black and white creative drawings, which is
what I knew he was trying to promote on the internet to sell
his artwork. I responded that Dorothy and I had enjoyed
coloring in mandalas when we were vacationing in the
Caribbean, but I made no mention about buying any of his
artwork, as I had no interest or intention of doing so. And this
is when our immediate back and forth e-mail communications
have come to an end.
Basically I am relieved at this break in the action between me
and Richie, as it felt like it was too much day-to-day
continuous contact with him. But it also feels rather strange
to me if there is legitimacy to my feeling that this break in the
action was directly related to me not responding to his hope
that I would be interested in buying his artwork on the
internet. It makes me wonder how genuine our friendship
truly is. But in all fairness to Richie, I should at least give him
the benefit of the doubt and candidly ask him if he was hurt
or disappointed about this and if this is why he stopped
sending me e-mails. At any rate, this is now nothing more than
a personal aside about the relationship I am having with an
old friend from Brooklyn, and it does not change anything in
regard to my evaluation if Divine Light Mission/Elan Vital and
Guru Maharajji/Prem Pal Rawat.
Conclusion
I am not at all surprised to learn from information and articles
on the internet, in particular from the Rick Ross Cult
Education site and the International Cultic Studies
Association, that there has been continuous disillusionment
and concerns about the legitimacy and ethical practices of
Guru Maharaji/Prem Pal Rawat and Divine Light Mission/Elan
Vital for many years⁵. However, I believe that my very
personal account from my lifelong friendship with Richie adds
Pinnacle Sociology & Anthropology ISSN: 2360-959X Page 6
a rather unique dimension to these reports. I concluded in my
Modern Religions book (Benjamin, 2013) that Divine Light
Mission had a moderate degree of cult danger, and I still
believe that this is the case4. The accumulation of wealth on
the part of Guru Maharajji/Prem Pal Rawat at the expense of
his impoverished devotees is certainly cause enough for me
to be very concerned about the ethics of his guruship⁴. Giving
one’s life to this guru at the expense of one’s freedom of
thought, as I have described in the series of essays in this
article in regard to my friend Richie, is another concern I have
about the cult dangers of this organization. As I conveyed in
my
Modern Religions book (Benjamin, 2013), there are
certainly modern religious organizations, such as Scientology
and the Unification Church, both of which are still in existence
today, that I have far more cultic concerns about than I have
in regard to Elan Vital⁶. But the dramatic reduction in
proselytizing behavior about his religion and guru from my
friend Richie is not a reason for me to have any less cultic
concerns about Pram Pal Rawat and Elan Vital than i described
in my above 1977 essay. The traps remain the same as I wrote
about nearly 40 years ago, and I must strongly discourage
anyone from partaking of them.
Notes
1. See Chapters 1: pages 37-41 and Chapter 3: pages 198-212 of
my book
Modern Religions: An Experiential Analysis and Exposé
(Benjamin, 2013).
2. “Love and Magic” is the phrase which Richie and Linda has used
to refer to their music and to the name of their performance as
a musical duo.
3. See my book
The Creative Artist, Mental Disturbance, and Mental
Health
(Benjamin, 2014) and my article My Conception of
Integral (Benjamin, 2006).
4. See Chapter 1: pages 37-41 in my
Modern Religions book
(Benjamin, 2013).
5. See the following websites:
ht tp : / /www. e x – p r emi e . o r g /p ap e r s /Ev i t a l . h tm
http://www.cultnews.com/category/elanvital/divinelightm
ission
http://culteducation.com/group/1219-divine-lightmission.
html
http://www.icsahome.com/system/app/pages/search?sco
pe=search- site&q=divine+Light+Mission&offset=20
6. See Chapters 1, 3, and 4 of my modern religions book (Benjamin,
2013).
7. However, i have since learned that as one progresses in the
Unification Church organization, it is indeed made quite clear
that Reverend Moon is the second Messia. See for example
Underwood & Underwood (1979).
8. See my personal experiential essays in Chapter 5 of my
Modern
Religions
book (Benjamin, 2013), and my essay Life, Death,
Meaning, and Purpose
(Benjamin, 2014b).
9. See my essay
Moonies and Premies in my Modern Religions book
(Benjamin, 2013), pages 204-206.
10. See my essay
On Internal in my book The Creative Artist, Mental
Disturbance, and Mental Health
(Benjamin, 2014).
11. See for example the book Sacred Journeys:
The Conversion of
Young Americans to Divine Light Mission
(Downton, 1979).
12. The nature of God being within the individual as well as in the
world is very much a part of various modern and new age
spiritual interpretations of God, including Neopaganism, Avatar,
and Conversations with God. See my related essays in Chapters
1 and 2 of my
Modern Religions book (Benjamin, 2013), and
Huston Smith’s (1991) book
The World’s Religions.
13. My essay
Experiential Skepticism: An Exploration of Mediumship
and Life after Death
(Benjamin, 2015) is also available on my
website:
www.benjamin-philsopher.com
References
1. Benjamin, E. (2006).
My Conception of Integral. Retrieved from
www.integralworld.net
2. Benjamin, E. (2013).
Modern Religions: An Experiential Analysis
and Exposé. Winterport, ME: Natural Dimension Publications.
3. Benjamin, E. (2014a).
The Creative Artist, Mental Disturbance,
and Mental Health
. Winterport, ME: Natural Dimension
Publications.
4. Benjamin, E. (2014b).
Life, Death, Meaning, and Purpose.
Retrieved from
www.integralworld.net
5. Benjamin, E. (2015).
Experiential Skepticism: An Exploration of
Mediumship and Life after Death
. Paranthropology Journal, Vol.
6, Issue No. 2, retrieved from
http://www.paranthropologyjournal.weebly.com/freepdf.
html
6. Downton, J. (1979). Sacred Journeys:
The Conversion of Young
American to Divine Light Mission
. New York: Columbia University
Press.
7. Smith, H. (1991).
The World’s Religions. New York: HarperCollins.
8. Underwood, B., & Underwood, B. (1979).
Moonstruck : A Memoir
of My Life in a Cult
. New York: William Morrow & Co.
9. Wilber, K. (1995).
Sex, Ecology, Spirituality. Shambhala.
]]>Pinnacle Psychology
(ISSN: 2360-9508)
http:/www.pjpub.org
© Author(s) 2015. CC Attribution 3.0 License.
Research Article
Humanistic Antidotes for a Social Media Technology Addicted Society
Elliot Benjamin, Ph.D.
Accepted 30 August, 2015.
ABSTRACT
This article describes some of the serious growing concerns about the dangers of widespread social media technology addiction in U.S. society,
and virtually all over the world. These concerns are especially prominent regarding the excessive use of social.....]]>
Pinnacle Psychology
(ISSN: 2360-9508)
http:/www.pjpub.org
© Author(s) 2015. CC Attribution 3.0 License.
Research Article
Humanistic Antidotes for a Social Media Technology Addicted Society
Elliot Benjamin, Ph.D.
Accepted 30 August, 2015.
ABSTRACT
This article describes some of the serious growing concerns about the dangers of widespread social media technology addiction in U.S. society,
and virtually all over the world. These concerns are especially prominent regarding the excessive use of social media technology by young
people, as conveyed by the author through excerpts from various articles. Additionally, the author describes his own psychology teaching
experiences at a university in Maine that illustrates through small group class discussions the concerns students have about the excessive use
of social media technology, in the form of narrative research. Humanistic antidotes based upon Carl Rogers’ initial formulation of humanistic
psychology through engaging in genuine, caring, authentic relationships with others are given, in particular with college students in the context
of humanistic education, as practiced by the author. Inclusive of taking time to express one’s own deepest self inclinations, the author concludes
with a positive vision of how the benefits of extensive information and communication through social media technology can be maintained
while avoiding its negative dehumanizing addictive dangers .
Keywords:
humanistic psychology, social media technology addiction, humanistic education.
Introduction
Sometimes I wonder about how Carl Rogers (1961) and
Abraham Maslow (1962), the acknowledged founders of
humanistic psychology, or Rollo May (1969), who was a
predominant force in the promotion of existential psychology,
would respond to the extreme level of how much our entire
current U.S. society, including in particular young people, are
immersed in their social media technologies. There has been
an increasing alarm the past few years about the extent that
especially young people in our U.S. society-and virtually all
over the world-are becoming saturated with what has been
referred to as “social media addiction” (Addiction.com Staff,
2012; Anderson, 2015; Augenbraun, 2014; Barnes, 2015;
Golinkoff, 2015; Gordon, 2015; IANS, 2015; Johnson, 2015;
McNamee, 2014; Paddock, 2013; Whiteman, 2014).¹ There
has also been a growing movement to include a psychiatric
disorder with the title of Internet Addiction Disorder (IAD)
listed in the
Diagnostic and Statistical Manual of Mental
Disorders (DSM-5)
(APA, 2013). Although Internet Addiction
Disorder is still not officially listed as a psychiatric disorder
in the DSM-V, the more narrowly defined Internet Gaming
Addiction was added in May, 2013 (APA, 2013; Augenbraun,
2014).
I have seen firsthand how young people appear to be addicted
to their social media technology from my teaching psychology
at Husson University the past 4 years. A number of my Husson
University Human Growth & Development and Introduction
to Psychology students frequently expressed in their essays
and group discussions how people no longer know how to
interact socially with each other, and how they are much more
comfortable in their online Facebook, SnapChat, etc.
relationships than in actually “talking” to people face-to-face.
Yes a number of my students somehow “know” that there is
something very wrong here, but I do not think they
understand the full context of what has taken place in regard
to mental health; i.e. the full context in regard to the
interpersonal emphasis of having genuine, caring, authentic
dialogue and relationships that is the essence of humanistic
psychology as formulated by Carl Rogers (1961). From my
experiences a few years ago as a community mental health
worker, I focused upon how much this kind of authentic
dialogue and relationships are needed by young people, in
particular young people with mental health issues, in our
society (Benjamin, 2011a). However, I am afraid that we as a
society are quickly moving past the point of being able to
experience anything approaching this kind of humanistic
relating, and virtually all the authors of the above articles on
social media technology addiction share my fears.
To relate to people authentically and caringly, one needs to
relate to one’s self as well. As conveyed by the above authors
on social media technology addiction, there appears to be a
widespread technological addiction that is taking over our
whole society and much of our whole world. Media technology
is in itself extremely impressive and useful, with extensive
benefits to extend and expand information and
communication. However, I think it is essential that people
control technology, and not the other way around. What I see
all around me is the billionaire technology entrepreneurs and
marketers successfully marketing their latest social media
technology-especially to children and adolescents-to a point
where our kids are growing up “vicariously,” i.e. being glued
to their computer technology gadgets as their favorite way of
passing time. Everything nowadays is quick, spontaneous, and
“online.” Whatever happened to “existential depth or awe”?
(Schneider, 2004). Going back a little further in time, I wonder
what Nietzsche (1885/2012) or Kierkegaard (1843/2013)
would say if they were able to see our current modern social
media technology society.
In this article I will describe what I have learned in regard to
college students’ perceptions of some of the adverse effects
of social media, from my psychology teaching experiences
over the past few years at Husson University in Bangor, Maine.
Although I did not initially set out to study this phenomenon,
the accounts of some of my students’ experiences with the
disturbing addictive aspects of over-indulgence in social
media that they have expressed in class discussions have had
much impact upon me. I will relate some of these accounts
from my students in this article, in the context of narrative
research (Chase, 2005; Clandinin, & Connelly, 2000;
Polkinghorne, 1988; Bochner, 2014).
One of the humanistic antidotes I convey in this article, in the
small context of my own college-level psychology teaching
experiences, is simply to give my students the assignment of
“talking” to each other, in small group discussions and whole
class sharings related to the topics of the psychology courses
Corresponding Author:
Elliot Benjamin, Ph.D.
Email: ben496@prexar.com
Pinnacle Psychology (ISSN: 2360-9508) Page 2
I teach at Husson University. This aspect of “sitting and talking
to each other” may sound very simple and natural, and in
many ways it certainly is. However, given what I will be
describing as my perspective of the social media technology
addiction that so many young people are caught up in, I
believe it is a significant event to have these young people
actually put away their technology when they are in their
classrooms and engage with each other in face-to-face
personal/academic authentic discussions. It is this context of
authentic engagement with each other and the intertwining
of feelings and “felt concerns” with the intellect, that falls
under the umbrella of humanistic education (Rogers, 1961,
1969)², that I think is essential to cultivate and develop to
balance out the extreme social media technology world that
we are all now living in.
Qualitative Research Methods: Narrative Research and
Autoethnography
A significant focus of my article is on what I have learned
about social media technology addiction from the informal
reports conveyed by some of my Husson University
psychology students in their small group class discussions.
Consequently the dominant research methods utilized in this
article is in the context of qualitative inquiry; in particular
narrative research and autoethnography (Bochner, 2014;
Chang, 2008; Chase, 2005; Denzin, 2014; Ellis, 2004, 2009;
Clandinin & Connelly, 2000; Jones, Adams, & Ellis, 2013;
Polkinghorne, 1988; Robson, 2002; Wertz, 2011). As I
mentioned in the Introduction, I did not initially set out to
study this topic and therefore this is by no means a formal
academic study. However, there is much overlap between
what I learned from my students in their narrative reports,
the narrative articles I have read on social media addiction,
and the basic ingredients of narrative and autoethnographic
research.
Autoethnography was developed in the last few decades of
the 20
century, largely through the efforts of sociologist
Carolyn Ellis (2004, 2009), and focuses upon the social
dynamics and context that the researcher is investigating.
However, unlike strict ethnographic research that does not
include personal reflections of the researcher,
autoethnography extends participant observation research
through placing a significant reliance upon the feelings,
thoughts, perspectives, experiences, reflections, insights, and
personal stories of the researcher, and often involves a high
level of personal vulnerability in terms of revealing
emotional/private aspects of oneself (Benjamin, 2013;
Bochner, 2014; Chang, 2008; Denzin, 2014; Ellis, 2004, 2009;
Jones, Adams, & Ellis, 2013). Thus in this article I will be
describing my personal reflections and related experiences
in the context of the narrative reports of my Husson University
students regarding their experiences and perspectives
concerning our society’s excessive immersion in social media
technology. However, I will begin by briefly describing a
recent personal experience I had with my significant other at
a local Maine restaurant that I believe has some bearing on
the social media technology issue that I am currently writing
about.
Relevant Personal Experience at a Restaurant
A few months ago my significant other and I were having
dinner at a local truck stop/family diner restaurant. The place
was quite crowded with long tables of people noisily talking
to each other, and it took a long time for our waitress to come
over to us. My significant other in particular did not appreciate
how noisy it was, and we were both mildly annoyed at how
slow the service was. However, I then looked around me at all
the people busily chatting away, and all of a sudden I realized
that there did not appear to be one cell phone being used by
any customer in the entire restaurant! Now this is 2015, and
the restaurant is in the vicinity of Bangor, Maine-a fairly large
city by Maine standards. I was especially sensitive to our
society’s generic cell phone environment from my teaching at
Husson University in Bangor. Nearly all the students at Husson
are continuously glued to their cell phones wherever they
are-in their classrooms when they are able to (including
surreptitiously using them inside their desk tables), in the
cafeteria, walking to and from classes, etc.
After I conveyed my observations to my significant other, we
both viewed all the noisy chatter in the restaurant in a
different light. For we now realized that these people were
actually “talking” to each other and enjoying being with each
other. It felt like some kind of uncanny throwback to an earlier
time before our modern computer age, when people actually
conversed with each other instead of sitting next to each other
and conversing on their social media cell phones. We no
longer minded all the noisy chatter in the restaurant, and our
waitress came over to us soon enough and apologized for the
long delay.
College Education and Social Media Immersion
In my Husson University Human Growth & Development and
Introduction to Psychology classes, I have increasingly
structured my classes to maximize the amount of time my
students are required to “talk” to each other, in small group
discussions and whole class sharings. After a while this
became fairly natural to my students, but in retrospect I can
appreciate the atmosphere I created in my classroom in a
similar way to how I appreciated my restaurant observations,
as I described above. My students were required to put away
their cell phones in my classroom (though I have no doubt
that some of them managed to still surreptitiously use them)
and literally as soon as class ended- as they were walking out
the door-many of them were back in their online worlds.
As I initially entered the classroom, at least half of my students
were busily engaged on their cell phones, and I always would
begin class by telling them to put away their recreational
technology devices. However, generally for at least one class
each week my students would talk to each other, in the context
of engaging in structured small group discussions, many of
which were quite personally engaging and revealing.
If this were the 1970s, when I was my students’ age and in
college, none of this would even be worth mentioning. For it
was the sign of the times to “talk” to each other and question
anything and everything about our society. Young people were
immersed in everything from antiwar demonstrations to
psychedelic drugs to free sex to women’s liberation to Eastern
religions (Taylor, 1999). But today young people are,
Pinnacle Psychology (ISSN: 2360-9508) Page 3
generally speaking, much more focused on effectively taking
their place in our extremely materialistic society. One striking
research example of this is evidenced by the significant
increase in freshmen college students, compared to a few
decades ago, who rate being well off financially as extremely
important and the significant decrease in freshmen college
students who rate having a meaningful philosophy of life as
extremely important (Pryor
et. al, 2007). However, there are
always exceptions to the rule, and one recent noteworthy
exception is the Occupy movement, where for a relatively
short period of time it felt to me like it was the 1970s again
and young people “woke up” (Benjamin, 2011b, Chomsky,
2013).
“THE” Psychology and Technological Addiction
The status quo is quite powerful, and the forces that I believe
are entrapping people now from cradle to grave are nothing
less than what I consider to be our whole technologically
addicted society and essentially world. But how can I have the
gumption to think that I am right about our whole society
suffering from technological addiction? Well I “feel” this when
I am doing my pure mathematics-working on discovering
some new truth in the sublime world of “my friends that don’t
exist.”³ I also feel this when I am working like the devil to
master my piano songs to successfully accompany the singers
in my theatre group, or to play the piano part of my Beethoven
violin/piano sonata well enough to return for a week this
summer to my very intensive adult chamber music school⁴.
And I feel this right now as I am writing this essay. For if I were
addicted to continuously engaging in social media interactions
on Facebook or elsewhere in the cyber world, this would
gradually replace the self immersion time I need to give myself
in order to continue the development in my above three
creative worlds. And if I were addicted to continuously
engaging in social media interactions on Facebook, I believe
this would also negatively infringe upon my ability to have
quality time and continue to engage in my harmonious,
intimate relationship with my significant other. Somehow my
genuine self would gradually wear away, and it would be
replaced by some kind of addictive personality who just could
not get enough of what online relationships with others could
offer it. But the scary part of all this is that I believe this is
essentially what is overtaking our whole society-with young
people at the forefront.
To feel-to care-to mull things over-to contemplate-to take
time-what happens to all these qualities when one is in the
whirlwind of spending virtually all of one’s free time either
surfing the web or in online interactions with people one has
never met? I contend that the essential qualities of both
humanistic and existential psychology are being lost as we
progress through the 21
century, in the midst of all it’s
staggering array of social media technology temptations. But
let me reaffirm-it is not the technology itself that is at fault.
It is the human temptation and potential for addiction, that is
being used in exceptionally skillful and effective ways by our
society’s technology expert marketers, that I believe is the
source of the problem. It feels like a runaway horse to me,
with no way of it being stopped. It goes along with the climate
debacle that spells disaster for the whole human race, once
again with no apparent way of being stopped (Gore, 2007).
The Occupy movement made a brave statement against
corporate greed (Benjamin, 2011b, Chomsky, 2013), and I
agree with the Occupy movement that it is precisely this greed
that has caused many of the ills of our society. Essentially the
greed that is the wheels of our super-capitalistic society serves
only the prospect of making more money-not the well-being
of our children or the propagation of the human species.
But if there is nothing that can be done about our
technologically addicted society, then why am I writing this
essay? Well-I am writing this essay because it is my basic
nature to express what I think and feel-which is how I
philosophize. I’ve mentioned how our social media technology
addicted society goes against the essential cores of humanistic
and existential psychology. I also believe it goes against the
essential core of transpersonal psychology. For transpersonal
psychology is engaged with the-for a lack of a better word-
“spiritual” component of what it means to be human
(Friedman & Hartelius, 2013). And I think that being
authentically “spiritual”-or engaging in deep self explorations
about life and existence-most definitely requires that one
spend extended time with one’s self, pondering all these
mysterious elements of life and the world. Engaging in
continuous online social interactions without taking sufficient
time to experience being alone with one’s own self on a
regular basis, does not go along with this kind of deep self
immersion that I think is at the core of transpersonal
psychology. I would like to coin the acronym THE psychology,
which stands for Transpersonal/ Humanistic/Existential
psychology. Using this acronym, one can summarize my main
argument in this essay by saying that our technological
addicted society is seriously undermining the ability of people
to engage in the essential cores of THE psychology. And from
my experiences of teaching psychology at the college level the
past few years, this undermining is extremely obvious in the
college classroom.
Anecdotal College Student Reports of Detrimental Aspects
of Social Media
In the various small group discussions that have taken place
in my Husson University psychology courses the past few
years, there have been a few communications that I think are
particularly revealing in regard to what can be construed as
the over-indulgence and/or misuse of social media
technology. To begin with, one of my Introduction to
Psychology students described someone in his high school
who was continuously engaged in online Facebook
communications, presenting the image of someone who was
exceedingly friendly, extroverted, and comfortable in various
social situations. However, my student was quite surprised
when he would see this same “online socially comfortable”
person walk down the school corridor every day with his head
down, avoiding any social contact with anyone and not even
responding when people would say hello to him. This opens
up the question of how could someone exhibit such
completely opposite personality characteristics when
interacting socially online with people compared to face-toface
social interactions? The answer was not difficult for my
students to come up with, as they easily described how one
puts on a façade or “persona” (Jung, 1961) when interacting
socially online, which can be quite different from one’s actual
personality when dealing with people face-to-face. Thus
someone can learn to be very “popular” online while becoming
Pinnacle Psychology (ISSN: 2360-9508) Page 4
less and less capable of relating to people face-to-face in any
kind of meaningful way.
This kind of example of contrasting online and face-to-face
personalities was repeated frequently in my Human Growth
& Development classes. One anecdotal report this past
semester that further reinforces this came from a student who
described her 11-year-old sister as being on her cell phone
virtually all the time, wherever she went and whatever she
did. This included any kind of social outing, family dinner,
athletic event, etc. But what particularly disturbed my student
was the complete lack of interest and ability her sister had in
being responsive to any face-to-face interaction that anyone
tried to engage her in. My student’s sister would answer
face-to-face questions with quick short Yes or No responses
and immediately get back to her much more interesting online
world in her cell phone. My student described how her
personal relationship with her sister was now virtually
nonexistent, and how her other family members felt the same
way.
And once again we can ask: What exactly is going on here? Is
“virtual reality” just another form of social reality? Although
some of my students would say this is the case, the majority
of them have become quite concerned that kids are growing
up these days with an overindulgence of social media
technology pervading every aspect of their lives.
Another example from my Human Growth & Development
class this past semester strikingly demonstrates the
phenomenon of people “hiding” in their online social media
worlds rather than “talking” to people who are actually
physically present. My student described the pattern of
challenging communications her roommate would engage her
in.
As it turns out, her roommate would wait for my student to
leave the room, and then immediately text her to convey
whatever particular problem she was having concerning my
student. This was her roommate’s way of avoiding the “messy”
personal interactions that face-to-face communication with
my student would involve, as it was so much easier and more
“efficient” to just communicate these things in a text message
since my student was not physically there to respond. Is there
anything wrong with this? There certainly is, according to my
student. It felt “inhuman” to her, and I must agree. Once again
we see the phenomenon of people bypassing face-to-face
relating, in particular when it involves some challenging
communications, in favor of the easier and relatively nonpersonal
form of an online text message.
The most recent example of what I have experienced as social
media technology addiction occurred when students in my
most past semester Husson Human Growth & Development
class were giving their end of semester class project
presentations.
As I mentioned above in a parenthetical remark, although I
always told my students at the beginning of class to put away
their recreational technology devices, I had no doubt that
some students were using their cell phones surreptitiously
inside their desks. In general this did not feel to me like it was
worth making an issue of during my class lectures, as it
involved just one or two students on an occasional basis,
although I did confront some students when I saw them doing
this when they were in their small group discussion. However,
when I was in the audience and observing two students
blatantly using their cell phones inside their desks during a
number of my students’ end of semester class presentations,
I knew I needed to deal with the situation. I wrote notes to
these students and made a general announcement about how
rude and disrespectful this was at the beginning of our next
class, which was the last class of the semester. This solved the
problem on the surface, as no student dared to surreptitiously
use his or her cell phone inside a desk during the class
presentation in our last class. But this whole situation made
quite the impression on me. For these same students who
made such inappropriate use of their cell phones during our
class previously had been quite vocal about the misuse of
social media technology during our small group discussion
related to this topic, quite recently. The last class presentation
was fittingly on the topic of the benefits and dangers of social
media technology, and I was glad that my class had ended with
this topic. But it struck me very forcefully how young people
in our society have become so far removed form anything
approaching Carl Rogers’ (1961) humanistic way of people
relating to each other.
There were a number of other examples described by many
students in these small group class discussions the past few
years that highlight the discrepancy between relating socially
online and face-to-face. Some of these examples strike me as
being quite bizarre, in spite of the fact that they are apparently
becoming increasingly more “normal” in our social media
technology society. For instance: families living under the
same roof and choosing to send each other text messages to
communicate rather than actually “talk” to each otherincluding
husbands and wives; students coming to class
before the instructor arrives and silently being immersed in
their social media world on their cell phones instead of talking
to each other; young children sitting immobile in front of their
computers to vicariously play their computer games or engage
in their online social media communications instead of getting
physical exercise playing with other children; college-age
people spending 12 hours a day immersed in their social
media technology, as disclosed by one of my Human Growth
& Development discussion groups. The list goes on and on,
and we haven’t even scratched the surface when it comes to
how all this social media technology immersion relates to
family interaction, childhood obesity, unstable financial
security through internet marketing, car accidents due to the
use of cell phones when driving, etc.
Excerpts from Research and Articles about Social Media
Technology Addiction
The descriptions I have given from some of my Husson
University small group discussions relating to the extreme
immersion of social media technology in our society are
actually quite similar to many of the experiences described in
the articles I listed above on social media addiction. Some
relevant excerpts of these articles are as follows¹.
Five years ago, Facebook was rarely mentioned in the context
of a marriage ending. But now it has become commonplace
for clients to cite social media use as a reason for divorce….A
recent study by Oxford University of 24,000 married
Pinnacle Psychology (ISSN: 2360-9508) Page 5
European couples found that the more they read about other
people’s exciting lives on social media, the more likely they
were to view their own with disappointment, leading to a
poorer sex life….So foreplay, emotional touch, fondling and
intense feelings count as much as a physical stimulant. With
smartphones in the bedroom, the emotional togetherness
that initiates sex is gone. (IANS, 2015, pp. 1-2)
A typical day for me starts out with me turning off the alarm
on my smartphone and immediately taking instant messages,
emails, status updates…you get the the picture. Every like,
every personal message, even the fun of interacting with
friends without having to actually, you know, be anywhere
near them….we’re so involved in “connecting” that we seem
to neglect really
connecting-far-reaching, pervasive effects in
the real world (that’s the stuff you see when you put your
phone away from pretty much everything)….We’re becoming
worse people; less rounded, more self-righteous, less
understanding and doing it to ourselves. (Gordon, 2015, p. 11)
You can be connected to the world around you twenty-four
hours a day, three hundred sixty-five days a year…. Over time
though, this connection can slowly become a need….Having
an online personality is easier for some people and their social
and emotional needs are quickly fulfilled. The amount of time
spent is rarely questioned and this is where the danger of a
possible addiction can come into play. (Addiction.com Staff,
2012, p. 6).
Their analysis showed that cell phone use was negatively
linked to GPA- the higher the cell phone use the lower the
grades-and positively linked to anxiety-higher cell phone use
was linked to higher anxiety. (Paddock, 2013, p. 2)
In our society, I occasionally feel as though we spend more
time documenting our time spent hanging out by trying to get
pictures and hashtags just right than we do actually talking
and connecting with one another. We are rarely ever fully
present in one place. We are out with our friends, but are more
focused on finding the best lighting for our selfies than
actually engaging in conversation. We are out to dinner, but
instead of savoring every flavor, we are compulsively making
sure to document it on SnapChat, Instagram, and Facebook.
We feel compelled to show up for our audience. We are
always connected, yet so fragmented. While becoming
increasingly connected we are becoming disconnected from
ourselves.
Because of the social media epidemic, it is also my observation
that time spent among friends is overshadowed by agendas.
It often feels contrived and staged. Social media breeds and
feeds shallow connections, and I find myself hungry for
deeper, more meaningful connections now more than ever.
We are starving ourselves while gorging on our incessant
compulsion to perform. Worst of all, it is a viciously
contagious epidemic….While at a coffee shop recently, I
observed two girlfriends who snapped at least 30 pictures, in
the span of about 30 minutes, all the while discussing their
social media lives. During their hour of time together, I lost
count of how many times I heard them reference Instagram,
SnapChat, and Twitter, for it dominated 80 percent of their
conversation. Most interestingly, they did not part ways
without one of them compulsively burying her face in her
phone and editing their new pictures. They were seated next
to each other in silence, one staring at the other as she edited
like a maniac. (Johnson, 2015, p. 12)
This is what I think is wrong with a social media obsessed
world. We can’t seem to enjoy something for what it is without
wanting everyone to know we’re enjoying it. We can’t dance
to the encore at our favorite band’s concert because we’re too
busy capturing it for our snap story, to make sure that people
know we’re doing something exciting. We can’t dig into a
beautiful meal before snapping pictures of it first, to make
sure that people know we are something delicious. We can’t
have a great hair day or wear something cute without taking
a photo and sharing it, making sure people know we looked
good that day. We can’t just be. (Barnes, 2015, p. 10)
Luo says long-established protocols for talking therapies for
addiction can be worthwhile components of a treatment plan
for Internet addiction. For example, he notes cognitive
therapy and motivational interviewing have been proven to
be especially useful for changing unhealthy behaviors….Luo
notes that potential treatments have to work in their own
cultural contexts, and what works for one group may not work
for another. The Daxing Internet Addiction Treatment
Center in Beijing, China, for example, treats addicted teens
with military-like exercise, and discipline, as well as
medication and other therapies, which may not go over so
well in other countries. (Augenbraun, 2014, pp. 3-4)
In India, the launch of the clinic appeared timely-in the same
week the Nimhans newspapers were reporting a case of a
13-year-old who hanged herself after her mother removed
her Facebook account….In Singapore, 87% of a population of
5.4 million own smartphones….China itself has over 300
Internet addiction centers. (McNamee, 2014, pp. 2, 4)
A related topic to what these excerpts powerfully convey as
our society’s dehumanizing social media technology addiction
that I always include in my required psychology small group
class discussions is something I will briefly discuss next:
media violence.
Media Violence, Human Caring/Sensitivity, and Real
World Violence
In an earlier article (Benjamin, 2012), I discussed the
relationship of playing violent video games and watching
media violence to real world violence. As I conveyed in my
article, although there is much alarming research that
demonstrates a high correlation between factors related to
media violence and real world violence, there is also research
that concludes there does not exist a significant correlation
between these factors (Benjamin, 2012). In both my Human
Growth & Development and Introduction to Psychology
classes, the vast majority of my students do not think that
playing violent video games or watching violence in the media
in general leads to real world violence, in spite of the alarming
research that their textbooks describe about this. What most
of my students think is that the factor of parental involvement
is the crucial issue; i.e. parents overseeing their children’s
watching violent television shows and playing violent video
games, and explaining to their children that these recreational
pastimes should not be reproduced in real life. I agree with
my students that the parent involvement factor is significant
in regard to the connection between media violence and real
Pinnacle Psychology (ISSN: 2360-9508) Page 6
world violence, and I also have conveyed to them that the
factor of mental disturbance is likely a very significant factor
in regard to the above high correlation (Benjamin, 2012).
However, I believe that the issue here goes much deeper and
is related to the growing alarm that our society is becoming
addicted to social media technology.
As many of my students expressed their concerns during our
social media technology small group discussions, as described
above, people nowadays “hide” behind their online Facebook,
SnapChat, etc. social images, and are losing the ability to
related to people face-to-face in personal and social ways
(Barnes, 2015; Gordon, 2015; Johnson, 2015). As one of my
Introduction to Psychology online CALCampus⁵ students
described in her excellent end of semester paper, the factors
of being personally detached from people and
watching/participating in so much vicarious media violence
may very well lessen the bonds between “make-believe”
violence and real world violence. And I agree with my online
student about this. The further removed our society becomes
from Carl Rogers’ humanistic visions of people personally,
caringly, and genuinely relating to each other, the easier it is
for at least some people-and it may very well be especially
some people with mental health disturbances-to commit real
world violence. For the online images of people are being
transformed from the real flesh and blood human beings that
we all are, to nothing more than quick and spontaneous
“virtual” interactions that are becoming increasingly devoid
of any of the ingredients of THE (transpersonal, humanistic,
existential) psychology that relate to depth and meaning of
life.
My Personal Philosophy to Overcome Temptation of
Technological Addiction
What are my own personal antidotes for living in this
whirlwind instantaneous information overload technology
society of ours while trying to retain my own essence? My own
personal antidotes are the same as they virtually always have
been-I absolutely need to continue doing what makes me who
I am. This means I continue to do my own mathematics, play
the piano, and philosophize (Benjamin, 2006). My three
worlds of mathematics, piano, and psychology/philosophy
have increasingly grown and extended from when I initially
described them as part of my “natural dimension” (Benjamin,
2013). But this is a good thing, and I am happy to see myself
extended through my mathematics publications, playing the
piano in chamber music groups and with a theatre company,
and publishing various essays in psychology and philosophy.
And the balance to my own creative self development and
pursuits is my harmonious intimate relationship with my
significant other for over 10 years, which is also an essential
part of what I originally envisioned as my “natural dimension”
(Benjamin, 2013). This intimate relationship context for me
is likewise developing and extending, as we have recently
bought a house together and plan on getting married next
year. And these are my own personal remedies for living in
what I, in agreement with R. D. Laing (1967) and Eric Fromm
(1955), consider to be our insane society (Benjamin, 2013).
Conclusion
In conclusion, it is apparent that social media technology is a
double-edged sword. Undoubtedly there are many significant
advantages of being able to communicate and obtain
information about virtually anything in the world very rapidly.
However, there are alarming growing concerns about the
harmful addictive aspects of this pervasive technology, as can
be seen in the sample of excerpts from articles I have included
above, from the above descriptions I have given of my own
teaching experiences at a university in Maine, and from the
movement to include technology addiction as a diagnostic
category by the American Psychiatric Association to
supplement the Internet Gaming Disorder that was included
in 2013 in the
Diagnostic and Statistical Manual of Mental
Disorders
(APA, 2013; Augenbraun, 2014).
However, there are also humanistic antidotes that I believe
have the potential of at least minimizing the detrimental
addictive effects of excessive social media technology use.
The humanistic antidotes that I have described all fit into the
guiding basic principles of Carl Rogers’ humanistic
psychology; i.e. engaging in genuine, caring, authentic
relationships with others. More specifically, in the arena of
humanistic education this means putting intention into
ensuring that students engage with one another in face-to-face
personal/academic meaningful discussions. In the arena of
parenting-in particular parenting of teenagers and young
adults-this means finding ways of maintaining the essential
bonds of the parent/child relationship that involves two-way
communication with a focus on caring, listening, and
authenticity. And perhaps most important of all, it is essential
that one maintains one’s own authentic self; i.e. to engage
oneself with one’s deepest self expressions and yearnings, and
perhaps have a sense of what one’s “calling” in life is. I believe
that if these humanistic antidotes were put into place on a
widespread scale, then social media technology would not
become a dangerous dehumanizing addiction, but rather
would serve as a very useful device to rapidly acquire
information and enhance communication all over the world
without the accompanying detrimental effects that I have
described in this article. Put more simply, we would control
our social media technology, and not the other way around.
Notes
1. See my recent article
Do We Live in a Social Media Technology
Addicted Society?
(Benjamin, 2015), which includes the article
excerpts described in the present article.
2. Humanistic education is based upon the work of humanistic
psychologists, in particular Abraham Maslow (1969) and Carl
Rogers (1961, 1969), and involves empathy, caring about
students, and genuineness on the part of the learning facilitator.
The basic principles of humanistic education include choice and
control, felt concern, the whole person, self evaluation, and the
teacher as a facilitator. For more information about humanistic
education, see
http://en.wikipedia.org/wiki/Humanistic_education
3. See my essay
My Conception of Integral (Benjamin, 2006), and
my essay On a
Natural Dimension of Mathematics in my book
The Creative Artist, Mental Disturbance, and Mental Health
(Benjamin, 2013).
4. My adult chamber music school is Kneisel Hall in Blue Hill,
Maine; see
www.kneiselhall.org As it turned out, due to a death
in the family I did not participate in my adult chamber music
week this summer, but I am planning on doing so again next
summer.
5. For more information about CALCampus see
www.calcampus.edu
Pinnacle Psychology (ISSN: 2360-9508) Page 7
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]]>
On the 2-Class Field Tower Conjecture for Imaginary Quadratic Number Fields
with 2-Class group of Rank 4
by Elliot Benjamin, Ph.D. February, 2015
Abstract
We demonstrate the existence of infinitely many new imaginary quadratic number fields k with 2-class group Ck,2of rank 4 such that k has infinite 2-class field tower. In particular, we demonstrate the existence of.....]]>
On the 2-Class Field Tower Conjecture for Imaginary Quadratic Number Fields
with 2-Class group of Rank 4
by Elliot Benjamin, Ph.D. February, 2015
Abstract
We demonstrate the existence of infinitely many new imaginary quadratic number fields k with 2-class group C_{k,2}of rank 4 such that k has infinite 2-class field tower. In particular, we demonstrate the existence of new fields k as above when the 4-rank of the class group C_{k} is equal to 1 or 2, and infinitely many new fields k in the case that the 4-rank of C_{k} is equal to 1, exactly three negative prime discriminants divide the discriminant d_{k} of k, and d_{k} is not congruent to 4 mod 8. This lends support to the conjecture that all imaginary quadratic number fields k with C_{k,2}of rank 4 have infinite 2-class field tower.
Introduction
Given an algebraic number field k, we denote by k^{1} the Hilbert 2-class field of k, i.e. the maximal abelian unramified extension of k with degree a power of 2. For nonnegative integers n we define the Hilbert 2-class field k^{n} inductively as k^{0} = k and k^{n+1 }= (k^{n})^{1}. Denoting by С the containment symbol, we define k^{0} Сk^{1} Сk^{2} С … k^{n} С … to be the 2-class field tower of k.
We say that the tower is finite if k^{n} = k^{n+1} for some n, with length n if n is minimal, and infinite otherwise.
It is well known that if k is an imaginary quadratic number field, C_{k,2} is the 2-Sylow subgroup
of the class group C_{k} (in the wide sense) of k, and rank C_{k,2} is the dimension over F_{2} of C_{k}/C_{k}^{2} where F_{2} is the finite field with two elements, then if rank C_{k,2} is greater than or equal to 5,
then k has infinite 2-class field tower [5]. It is also well known that for k imaginary quadratic with rank C_{k,2 }= 2 or 3, then the 2-class field tower of k may be finite or infinite, and that if rank C_{k,2 }= 1 then the 2-class field tower of k is finite and has length 1(cf. [9], [11], [12], [17]). It was conjectured in the late 1970s that if k is imaginary quadratic with rank C_{k,2 }= 4, then k has infinite 2-class field tower (cf. [12], [13]). In 1996 and 2000 Hajir, extending previous work that Koch (1969) had done in a particular case, used simpler techniques than Koch to prove a partial result in this direction: namely that if k is an imaginary quadratic number field such that C_{k} has 4-rank of 3 or 4 then k has infinite 2-class field tower (cf. [6], [7] and Lemma 1 below). From our own work in the early 2000s we obtained further partial results in the direction of the 2-class field tower conjecture, focusing upon the number of negative prime discriminants dividing the discriminant d_{k} of k, the Kronecker symbols of the primes dividing d_{k}, the congruence class mod 8 of d_{k}, and the 4-rank of C_{k} (cf. [1], [2]). Subsequent to our own work, in the 2000s Sueyoshi used Rédei matrices to improve upon our results in the cases when exactly one negative prime discriminant divides d_{k} and when exactly three and five negative prime discriminants divide d_{k}, in particular proving that k has infinite 2-class field tower when k is such that the 4-rank of C_{k} is equal to 1, five negative prime discriminants divide d_{k}, and d_{k} is not congruent to 4 mod 8 (cf. [15], [16], [18], [19], [20]). And in 2010 Mouhib improved upon Sueyoshi’s results in the one negative prime discriminant case, proving that k always has infinite 2-class field tower if exactly one negative prime discriminant divides d_{k} (cf. Lemma 3 and [14]).
From the above we see that for the cases when exactly three or five negative prime discriminants divide d_{k},we, as well as Sueyoshi, have obtained partial results in the direction of the 2-class field tower conjecture (cf. [1], [2], [18], [19], [20] and Lemmas 4 through 7 below).
We summarize the above historical results for when we know the 2-class field conjecture is
satisfied as follows, where k is an imaginary quadratic number field with rank C_{k,2} = 4, “Negative Prime Discriminants” denotes the exact number of negative prime discriminants dividing d_{k}, “Congruency” denotes whether or not d_{k} is congruent to 4 mod 8, and “Literature” refers to the researcher who initially completely proved the final result.
Table 1: Cases when 2-Class Field Conjecture is Satisfied
4-rank of C_{k} Negative Prime Discrminants Congruency Literature
3 or 4 1, 3, or 5 no restrictions Hajir, 1996, 2000
0, 1, 2, 3, or 4 1 no restrictions Mouhib, 2010
2 5 no restrictions Benjamin, 2002
2 3 not 4 mod 8 Benjamin, 2002
1 5 not 4 mod 8 Sueyoshi, 2009
_{ } For our present purpose we define “new” imaginary quadratic number fields k with rank C_{k,2 }= 4 and infinite 2-class field tower, as fields that do not satisfy the conditions of any of the cases of Lemmas 4 through 7 below, and to the best of our knowledge have not been reported in the literature, either as a part of families or as separate examples. By applying a result by Schmithals (1980), we are able to demonstrate the existence of new imaginary quadratic number fields k with infinite 2-class field tower when rank C_{k,2 }= 4 and C_{k} has 4-rank 1 or 4-rank 2.
In particular, we show that there exist infinitely many new imaginary quadratic number fields k with infinite 2-class field tower when k is such that the 4-rank of C_{k} is equal to 1, exactly
three negative prime discriminants divide d_{k}, and d_{k} is not congruent to 4 mod 8. In the 4-rank 2 case, we obtain new fields k such that exactly three negative prime discriminants divide d_{k, }
d_{k} is congruent to 4 mod 8, and the Kronecker symbols of the primes dividing d_{k} satisfy the conditions we have previously given for which we do not know in general if the 2-class field tower is finite or infinite (cf. [2] and Case C of Lemma 5 below).
Preliminaries
We begin by stating the Golod & Shafarevich Inequality (as refined by Gaschutz and Vinberg), and two related inequalities that have been derived from a more generic inequality by
Martinet (as described in [19]), all of which have been quite useful in obtaining the results that
we give in Lemmas 3 through 7 (cf. [5], [10], [13], [19], [20]).
Lemma 1: Golod & Shafarevich Inequality (refined by Gaschutz and Vinberg): Let k be a number field, C_{k} be the class group of k, and E_{k} be the group of units of k. Then the 2-class field tower of k is infinite if rank C_{k,2} is greater than or equal to 2 + 2(√(rank_{2}(E_{k}) + 1)), where rank_{2}(E_{k}) is the dimension of the elementary 2-group E_{k}/E_{k}^{2}considered as a vector space over F_{2} (and can be described as the number of infinite primes of k).
Lemma 2: i) Let F be a totally real number field of degree n, and E be a totally imaginary quadratic extension of F. Let t be the number of prime ideals of F which ramify in E. If t ≥ 3 + 2√(n + 1), then the 2-class field tower of E is infinite.
ii) Let F be a totally imaginary number field of degree n, and E be a quadratic extension of F. Let t be the number of prime ideals of F which ramify in E. If t ≥ (n/2) + 3 + 2(√(n + 1)), then the 2-class field tower of E is infinite.
We next combine our aforementioned results by Hajir and Mouhib (cf. [5], [6], [14]) into the following lemma.
Lemma 3: Let k be an imaginary quadratic number field such that C_{k} has 4-rank ≥ 3, or
rank C_{k,2 }= 4 and exactly one negative prime discriminant divides d_{k}. Then k has infinite
2-class field tower.
For the remaining cases when exactly three or five negative prime discriminants divide d_{k}
for k an imaginary quadratic number field with rank C_{k,2 }= 4, we state our previous results
(cf. Lemmas 4, 5, and 6), as well as the relevant results of Sueyoshi (cf. Lemma 7), in the form
of the four lemmas given below (cf. [1], [2], [19], [20]). In Lemma 7 we utilize only parts ii)
and iv) in Sueyoshi’s Proposition in [19] and [20], as parts i) and iii) are contained in our own previous results (see cases A and B of Lemma 4 below). However, we first describe how we obtain the 4-ranks of our quadratic number fields.
To obtain the 4-ranks of the class groups of our various fields, we will utilize both
d_{k}-splittings of the second kind and Rédei matrices (cf. [15], [16], ]18], [19], [20]). Given an algebraic number field k, we define a d_{k}-splitting of the second kind to be a splitting of d_{k} into two relatively prime fundamental discriminants (d_{1}, d_{2}) such that the Kronecker symbols
(d_{1}/p) = 1 and (d_{2}/q) = 1 for all primes p dividing d_{2} and for all primes q dividing d_{1}, and where we consider the splitting (1, d_{1}d_{2}) to be the trivial d_{k}-splitting of the second kind. It is
well known that for any quadratic (number field k, if s is the number of d_{k}-splittings of the second kind and r is the 4-rank of the narrow-class group of k, then 2^{r} = s (cf. [16]).
We also use Rédei matrices, which are described as follows (cf. [16], [17], [19], [20], [21]), where k is a quadratic number field. Let d_{k} = p_{1}*p_{2}*…p_{t}* be the unique factorization of d_{k} into a product of prime discriminants, where p* = (-1)^{(p-1)/2)}p if p is an odd prime, and p* = -4, 8, or -8 if p = 2. The Rédei matrix R_{k} = (a_{ij}) with entries in the finite field F_{2}, is describedas a matrix consisting of Kronecker symbol representations a_{ij} such that -1 to the exponent a_{ij} equals {(p_{i}*/p_{j}) if i ≠ j, and ((d_{k}/p_{i}*)/p_{i}) if i = j}. For properties of Rédei matrices see the above references, and in particular we mention that the sum of all row vectors of R_{k} is equal to the zero vector in F_{2}, and the following 4-rank property of Rédei matrices that we will make much use of to establish our results in this paper.
Rédei matrix 4-rank property: The 4-rank of the narrow class group of a quadratic number field k is equal to t – 1 – rank R_{k}, where t is the number of prime discriminants dividing d_{k}, and if k is imaginary this translates into the 4-rank of C_{k} is equal to t – 1 – rank R_{k}.
We now state our four lemmas mentioned above.
Lemma 4: Let k be an imaginary quadratic number field such that rank C_{k,2 }= 4. Then k has infinite 2-class field tower in the following cases:
A) there exists a negative prime discriminant d_{j} dividing d_{k, }such that (-p_{j}/p_{i}) = 1 where p_{j} is the prime dividing d_{j}, all p_{i} are distinct, and p_{i} is distinct from p_{j} for all primes p_{i} dividing d_{k,}
1 ≤ i ≤ 5;
B) for some prime p_{j} congruent to 1 mod 4, or p_{j} = 2, in which case we further assume that 8 is a fundamental discriminant dividing d_{k}, we have (p_{j}/p_{k}) = (p_{k}/p_{l}) = (p_{j}/p_{m}) = 1, p_{j}, p_{k}, p_{l}, p_{m} distinct
primes dividing d_{k};
C) at least two of the prime discriminants dividing d_{k} are positive, and (p_{1}/p_{3}) = (p_{2}/p_{3}) = 1 where p_{1} and p_{2} are distinct primes dividing positive prime discriminants dividing d_{k}, and p_{3} is a prime dividing a positive or negative prime discriminant dividing d_{k}, where p_{3} is not equal to p_{1} or p_{2}.
Remark 1: We note that in Lemma 1 of [2] we mistakenly stated the requirement in Case B above as (p_{j}/p_{k}) = (p_{k}/p_{l}) = (p_{j}/p_{m}) (see Lemma 1 in [1] for the correct statement).
Lemma 5: Let k be an imaginary quadratic number field with rank C_{k,2 }= 4 and 4-rank of C_{k}
equal to 2. Then k has infinite 2-class field tower in each of the following cases:
A) five negative prime discriminants divide d_{k};
B) d_{k} not congruent to 4 mod 8;
C) d_{k} congruent to 4 mod 8 and exactly three negative prime discriminants divide d_{k},
and the Kronecker symbols of the primes dividing d_{k} do not have the format (p_{1}/q_{1}) = (p_{1}/q_{2}) = (p_{1}/q_{3}) = (p_{2}/q_{1}) = (p_{2}/q_{2}) = (p_{2}/q_{3}) = -1, where p_{1} and p_{2} are distinct primes dividing positive prime discriminants dividing d_{k}, and q_{1}, q_{2}, and q_{3} are distinct primes dividing negative prime discriminants dividing d_{k}._{ }
For our next lemma, we note that If k is an imaginary quadratic number field with 4-rank 1, then from our above description of d_{k}-splittings of the second kind we see that there is a unique non-trivial d_{k}-splitting of the second kind for k. The following lemma is a corrected version of Lemma 6 in [2] from our earlier work (see Remark 2 below), in which we use the following notation: p_{1}^{*} and p_{2}^{*} are distinct positive prime discriminants dividing d_{k}, q_{1}^{*} is a negative prime discriminant dividing d_{k}, and r_{i}^{*}, 2 ≤ i ≤ 5, represent distinct positive prime discriminants or negative prime discriminants dividing d_{k} such that r_{i}^{*} is distinct from p_{1}^{*}, p_{2}^{*}, and q_{1}^{*}.
Lemma 6: Let k be an imaginary quadratic number field with rank C_{k,2 }= 4 such that C_{k} has 4-rank 1. If the nontrivial d_{k}-splitting of the second kind is either (without loss of generality) (q_{1}^{*}, r_{2}^{*}r_{3}^{*}r_{4}^{*}r_{5}^{*}) or (p_{1}^{*}, r_{2}^{*}r_{3}^{*}r_{4}^{*}r_{5}^{*}), or if d_{k} is not congruent to 4 mod 8 and the nontrivial d_{k}-splitting of the second kind is (p_{1}^{*}p_{2}^{*}, r_{1}^{*}r_{2}^{*}r_{3}^{*}), then k has infinite 2-class field tower.
Remark 2: We note that in the statement of Lemma 6 in [2] we mistakenly included the case
d_{k} congruent to 4 mod 8 with the nontrivial d_{k}-splitting of the second kind (p_{1}^{*}p_{2}^{*}, r_{1}^{*}r_{2}^{*}r_{3}^{*}).
For parts i and ii of the following lemma, cf. [19], [20]: Proposition: parts ii and iv.
Lemma 7: Let k be an imaginary quadratic number field and let F be subfield of the genus field of k.
i) Suppose that F/Q is a totally real biquadratic extension, where Q is the field of rational numbers. If three rational primes split completely in F and another rational prime is unramified in F and these four rational primes ramify in k, then the 2-class field tower of k is infinite.
ii) Suppose that F/Q is a totally imaginary biquadratic extension. If two rational primes split completely in F and another rational prime is unramified in F and these three ramified primes
ramify in k, then the 2-class field tower of k is infinite.
iii) Let k be an imaginary quadratic number field such that rank C_{k,2 }= 4, C_{k} has 4-rank 1,
five negative prime discriminants divide d_{k}, and d_{k} is not congruent to 4 mod 8. Then k has infinite 2-class field tower, where Q is the field of rational numbers.
Remark 3: Recall the well-known result from genus theory (cf. [3], [8], [16]) that if k is a quadratic number field with discriminant d_{k} and t is the number of primes that ramify in k (which is the number of primes that divide d_{k}), then rank C_{k,2} = t – 2 if d_{k} > 0 and is not a sum of two squares, and rank C_{k,2} = t – 1 otherwise.
We also state a result from Mouhib (2010), that he utilized to demonstrate that there are
imaginary quadratic number fields k with rank C_{k,2 }= 2 and 4-rank of C_{k} equal to 2 such that
k has infinite 2-class field tower (see Remark 11 in the final section). We discuss this result in regard to an open question about the 2-class field tower conjecture (see Question 4 in the Open Questions section).
Lemma 8: Let d be a square-free positive integer such that d is not congruent to 1 mod 4,
let M = Q(√d), and assume that 8 divides C_{M}. Then for every prime number q congruent to
3 mod 4 such that the equation x^{2} – dy^{2} = q has a solution in Z x Z (where Z is the ring of
integers), the imaginary quadratic number field Q(√-qd) has infinite 2-class field tower.
Remark 4: We note that given the assumptions of Lemma 8, it must be the case that
x and y are both odd, and d is congruent to 2 mod 4.
An Application of a Result by Schmithals
to Obtain New Imaginary Quadratic Number Fields k with Rank C_{k,2 }= 4
that Satisfy the 2-Class Field Tower Conjecture
To find new fields k that satisfy the 2-class field tower conjecture we now utilize the following application of a result by Schmithals (cf. the inequalities (1) and (3) in [17]) where k is an imaginary quadratic number field, F is a real quadratic number field, K is the compositum of
k^{1} and F, h denotes the 2-class number (in the wide sense) of F, and m is the number of primes q dividing d_{k} such that (d_{F}/q) = -1.
Lemma 9: If m ≥ (1/h)(3 + 2√(2h + 1) then k has infinite 2-class field tower.
Remark 5: Lemma 9 follows immediately from Schmithals’ more generic inequalities in [17] (cf. inequalities (1) and (3)). As noted by Schmithals, a more generic form of Lemma 9
(cf. inequality (3) in [17]) is satisfied with m = 1 and h ≥ 16.
Remark 6: Schmithals utilized what we have referred to as Lemma 9 to prove that there are infinitely many imaginary quadratic number fields k with exactly three primes dividing d_{k}
that have infinite 2-class field tower (cf. Satz 1 and Beispiel 4, which is k = Q(√-5.11.461),
in [17]). Utilizing our above Remark 3 and 4-rank of C_{k} criteria, we see that in effect Schmithals proved in 1980 that there are infinitely many imaginary quadratic number fields k with
rank C_{k,2 }= 2 and 4-rank of C_{k} equal to 2 that have infinite 2-class field tower. Mouhib in 2010 proved this same result (cf. Prop. 3.3 in [14]), referring to Schmithals’ example
k = Q(√5.11.461), but he did not mention Schmithals’ essential proof of the result. For historical accuracy we would like to emphasize here that the proof of this result should be attributed to Schmithals in 1980.
Utilizing Lemma 9 we immediately obtain the following lemma, which is what we utilize to obtain our new imaginary quadratic number fields k with rank C_{k,2 }= 4 and infinite 2-class field tower.
Lemma 10: Let p_{i}, q_{i}, i = 1, 2, 3, 4, be distinct prime numbers such that p_{i} is congruent to 1 mod 4 and q_{i} is congruent to 3 mod 4, and let q_{5} be a prime number such that q_{5} is congruent to
3 mod 4, or q_{5} = 2 if q_{5}* = -4 or q_{5}* = -8, and q_{5} is not equal to q_{i} for i = 1, 2, 3 or 4. Without loss of generality let M = Q(√p_{1}p_{2}q_{1}) (resp. Q√q_{1}q_{2}q_{3}), Q√q_{1}q_{2}q_{3}q_{4}), Q(√p_{1}p_{2}q_{1}q_{2}), Q√2q_{1}q_{2}q_{3}), Q(√2p_{1}q_{1}q_{2}), Q(√2p_{1}p_{2}q_{1}), Q(√2p_{1}p_{2}p_{3}), Q(√p_{1}p_{2}p_{3}p_{4})). Assume that 16 divides h(M), and (4p_{1}p_{2}q_{1}/q_{5}) = -1 (resp. (4q_{1}q_{2}q_{3}/q_{5}) = -1, (q_{1}q_{2}q_{3}q_{4}/q_{5}) = -1, (p_{1}p_{2}q_{1}q_{2}/q_{5}) = -1, (2q_{1}q_{2}q_{3}/q_{5}) = -1, (2p_{1}q_{1}q_{2}/q_{5})= -1, (2p_{1}p_{2}q_{1}/q_{5}) = -1, (2p_{1}p_{2}p_{3}/q_{5}) = -1, (p_{1}p_{2}p_{3}p_{4}/q_{5}) = -1). Let L be an imaginary quadratic number field with exactly five primes dividing d_{L}, and moreover let L = Q(√-p_{1}p_{2}q_{1}q_{5}) (resp. Q√-q_{1}q_{2}q_{3}q_{5}), Q√-q_{1}q_{2}q_{3}q_{4}q_{5}), Q√-q_{1}q_{2}q_{3}q_{4}), Q(√-p_{1}p_{2}q_{1}q_{2}q_{5}), Q(√-p_{1}p_{2}q_{1}q_{2}), Q√-2q_{1}q_{2}q_{3}q_{5}), Q(√-2p_{1}q_{1}q_{2}q_{5}), Q(√-2p_{1}p_{2}q_{1}q_{5}), Q(√-2p_{1}p_{2}p_{3}q_{5}), Q(√-2p_{1}p_{2}p_{3}p_{4}), Q(√-p_{1}p_{2}p_{3}p_{4}q_{5}), Q(√-p_{1}p_{2}p_{3}p_{4})). Then L has infinite 2-class field tower.
In the next section we will demonstrate by the use of Lemma 10 the existence of some new imaginary quadratic number fields k such that rank C_{k,2 }= 4 and C_{k} has 4-rank 2, which satisfy the 2-class field tower conjecture as described above. For the new fields that we will obtain, the Kronecker symbols listed in Case C of Lemma 5 all have value -1, and these fields satisfy the discriminant conditions of Lemma 5; in particular d_{k} is congruent to 4 mod 8, exactly three negative prime discriminants divide d_{k}, and the Kronecker symbols of the primes dividing d_{k} have the format (p_{1}/q_{1}) = (p_{1}/q_{2}) = (p_{1}/q_{3}) = (p_{2}/q_{1}) = (p_{2}/q_{2}) = (p_{2}/q_{3}) = -1, where p_{1}, p_{2}, q_{1}, q_{2}, and q_{3} are defined as in Case C of Lemma 5. To obtain these new fields, we make use of Rédei matrices as we have described above (cf. [18], [19], [20]), and we give a particular formulation in the context of the following lemma.
Lemma 11: Let k be an imaginary quadratic number field such that rank C_{k,2 }= 4 and C_{k} has
4-rank 2, exactly three negative prime discriminants divide d_{k}, d_{k} is congruent to 4 mod 8, and the Kronecker symbols of the primes dividing d_{k} satisfy the equalities listed in Case C of
Lemma 5. Without loss of generality let q_{1} = 2, and M_{1}, M_{2}, and M_{3} denote the corresponding real quadratic fields M_{1} = Q(√p_{1}p_{2}q_{2}q_{3}), M_{2} = Q(√p_{1}p_{2}q_{2}), and M_{3} = Q(√p_{1}p_{2}q_{3}). Then
h(M_{2}) = h(M_{3}) = 4, and h(M_{1}) ≥ 8; h(M_{1}) ≥ 16 if and only if the narrow class group of M_{1} has
8-rank 1. If (p_{1}/p_{2}) = 1 and the narrow class group of M has 8-rank 1, then k is a new field with infinite 2-class field tower.
Proof: We see from Sueyoshi (cf. [19], [20]) that the Rédei matrices of fields k that satisfy the assumptions of the lemma can be described as follows, where the entries a_{ij} are in the field F_{2}, 1 ≤ i ≤ 5, 1 ≤ j ≤ 5; if i ≠ j then a_{ij} = 1 if and only if (r_{i}/r_{j}) = -1, and if i = j then a_{ij} = 1 if and only if [d_{k}/r_{i}^{*}]/r_{i}) = -1, where r_{i},1 ≤ i ≤ 5, are the primes dividing d_{k}, r_{1}^{*} = -4, r_{1} = q_{1} = 2, r_{2}^{*} < 0,
r_{3}^{*}< 0, r_{4}^{*} > 0, r_{5}^{*} > 0, and [d_{k}/r_{i}^{*}] denotes d_{k} divided by the prime discriminant r_{i}^{*}; the symbol * is used to denote that a_{ij} can have the value 0 or 1, with the stipulations described below; we denote r_{2} = q_{2}, r_{3} = q_{3}, r_{4} = p_{1}, and r_{5} = p_{2}.
Then R_{k} = [* 1 1 0 0
* 1 1 1 1
* 0 0 1 1
1 1 1 * *
1 1 1 * *]
As described by Sueyoshi (cf. [19]), it is understood that this Rédei matrix formulation of R_{k} can have a change of order of the r_{i}’s, 2 ≤ i ≤ 5, the sum of all row vectors of R_{k} is equal to the zero vector in F_{2}, and if a_{44} = 0 then a_{45} = a_{54} = a_{55} = 0 and a_{11} = 1, and if a_{44} = 1 then a_{45} = a_{54} = a_{55} = 1 and a_{21} = 1; consequently there are four generic types of these Rédei matrices. From an examination of the matrices of M_{1}, M_{2}, and M_{3}, we see that the ranks of these matrices are respectively 2, 3, and 3, and from Remark 3 and the 4-rank formula for the narrow class group using Rédei matrices (see Preliminaries), along with the well-known fact that if a prime congruent to 3 mod 4 divides d_{k }then the rank of the narrow group is twice the rank of the wide class group (see for example [16]), we obtain that the 2-class groups of M_{2} and M_{3} are elementary (i.e. have 4-rank 0), and the 2-class group of M_{1} has 4-rank 1, which implies that h(M_{2}) = h(M_{3}) = 4 and h(M_{1}) ≥ 8. It is immediate from what we have described above that the condition h(M_{1}) ≥ 16 is equivalent to the narrow class group of M_{1} having 8-rank 1, which is equivalent to the (wide) class group of M_{1} also having 8-rank 1, and there is a known criteria using the Hilbert symbol to determine the 8-rank of a quadratic number field (cf. [21]). Finally, if (p_{1}/p_{2}) = 1 (meaning that (r_{4}/r_{5}) = 1) we see from the description of R_{k} given above that a_{11} = 1 and consequently we must have a_{21} ≠ a_{31}. Therefore we obtain the Kronecker symbol (p_{1}p_{2}q_{2}q_{3}/2) = -1, which along with the assumption that the narrow class group of M_{1} has 8-rank 1 (which as we have seen, is equivalent to h(M_{1}) ≥ 16) gives us the required conditions of Lemma 10 to obtain that k has infinite 2-class field tower. Since we are not able to utilize either Lemma 4 or Lemma 7 (or any other formulations in the literature to the best of our knowledge; see Case 60 in [20]) to show that k has infinite 2-class field tower, k is a “new” field and our lemma is established.
NOTE: For the remainder of this paper, unless stated otherwise k will always denote an imaginary quadratic number field with rank C_{k,2 }= 4. _{ }
The following lemma is useful to show that if a field L satisfies the conditions of Lemma 11, then there are infinitely many such fields that also satisfy these conditions, and consequently have infinite 2-class field tower, which we will make use of to show that there are infinitely many new fields k that satisfy the 2-class field tower conjecture when C_{k} has 4-rank 1.
Lemma 12: Assume there exists a field L that satisfies the conditions of Lemma 10. Then there exist infinitely many such fields that also satisfy these conditions, and therefore there exist infinitely many such fields that have infinite 2-class field tower that satisfy the conditions of Lemma 10.
Proof: By way of illustration, we demonstrate the result for two specific cases of Lemma 10, using the Chinese Remainder Theorem (CRT) and Dirichlet’s Theorem of Primes in an Arithmetic Progression (DPAP). Let M = Q(√p_{1}p_{2}q_{1}) and L = Q(√-p_{1}p_{2}q_{1}q_{5}). Using CRT and DPAP we are able to formulate infinitely many fields L of the form Q(√-p_{1}p_{2}q_{1}q) where q is a prime congruent to 3 mod 4 and (p_{1}/q_{5}) = (p_{1}/q), (p_{2}/q_{5}) = (p_{2}/q), and (q_{1}/q_{5}) = (q_{1}/q). Now let M = Q(√p_{1}p_{2}q_{1}q_{2}) and L = Q(√-p_{1}p_{2}q_{1}q_{2}). Similarly, we once again use CRT and DPAP, to formulate infinitely many fields L of the form Q(√-p_{1}p_{2}q_{1}q_{2}q) where q is a prime congruent to 3 mod 4, and (p_{1}/2) = (p_{1}/q), (p_{2}/2) = (p_{2}/q), (-q_{1}/2) = (2/q_{1}) = (q_{1}/q), and (-q_{2}/q) = (2/q_{2}) = (q_{2}/q). Since (p_{1}p_{2}q_{1}q_{2}/2) = (p_{1}p_{2}q_{1}q_{2}/q), we see that we again have obtained infinitely many fields that satisfy the conditions of Lemma 10, and consequently have infinite 2-class field tower. The remaining cases of Lemma 10 are done similarly, and we leave the details to the reader.
Remark 7: Our use of CRT and DPAP in the proof of Lemma 12 generalizes the technique Mouhib used in the rank C_{k,2 }= 2 case, as described in the remark following Prop. 3.3 in [14] (however, see Remark 6 above). Note that if k is a field described in Lemma 11 for which we are able to show has infinite 2-class fields tower by means of Lemma 10, then we must have M_{1} = Q(√p_{1}p_{2}q_{2}q_{3}). Therefore when we use CRT and DPAP as described in the proof of Lemma 12, the infinitely many fields we obtain all have d_{k} not congruent to 4 mod 8, and therefore by Lemma 5: Part b we know that any such field k in this infinite collection for which C_{k} has
4-rank 2 is not a new field.
We now demonstrate the existence of new fields k with infinite 2-class field tower such that
rank C_{k,2 }= 4 and C_{k} has 4-rank 2, three negative prime discriminants divide d_{k}, d_{k} is congruent to
4 mod 8, and the Kronecker symbols of the primes dividing d_{k} satisfy the equalities listed in
Case C of Lemma 5.
Case 1: 4-Rank of C_{k} Equal to 2
We see from above that in the 4-rank of C_{k} equal to 2 case we know that k has infinite 2-class field tower except for one family of fields, and from [2] we see that this is specifically the family of fields such that d_{k} is congruent to 4 mod 8, exactly three negative prime discriminants divide d_{k}, and the Kronecker symbols of the primes dividing d_{k} have the format (p_{1}/q_{1}) = (p_{1}/q_{2}) = (p_{2}/q_{2}) = (p_{2}/q_{2}) = (p_{1}/2) = (p_{2}/2) = -1, where p_{1} and p_{2} are primes congruent to 1 mod 4 dividing d_{k}, and q_{1} and q_{2} are primes congruent to 3 mod 4 dividing d_{k}. To make use of Lemma 11 to obtain our new fields, we begin by defining a Rédei sub-type matrix of 4-rank n, n = 0, 1, or 2, which we denote by R_{n}, to be a matrix with particular conditions that is contained in a family S of generic Rédei matrices listed in [19] or [20], such that if k is a field with the property that the primes dividing d_{k} satisfy the Kronecker symbol matrix entries of R_{n} (which we will denote by saying that “k satisfies R_{n}”), then C_{k} has 4-rank n and k does not satisfy any matrix in S – R_{n}. Once again the matrix entry * signifies a choice of 0 or 1, and when d_{k} is congruent to 4 mod 8 it is understood that q_{1}* = -4. It is also understood that these Rédei matrices are generic in the sense that aside from q_{1}* = -4, the order of the primes dividing positive prime discriminants (resp. dividing negative prime discriminants) can be changed (cf. [18]. [19], [20]). We refer to a particular Rédei sub-type matrix R_{n} as “open” if there exists
a field k as above that satisfies R_{n} for which it is not known if k is a new field with infinite
2-class field tower.
From Case 60 in [20] we see that there are exactly four open Rédei sub-type matrices of 4-rank 2, and that if k satisfies any one of these four matrices then k has the above Kronecker symbol format. We refer to fields k with 4-rank 2 that have the above Kronecker symbol format as Family D_{2} fields. From [20] we see that a_{23} = a_{41 }=a_{51 }= a_{24 }= a_{25 }= a_{34 }=a_{35 }=1 for all four of these open Rédei sub-type matrices, and that we can distinguish these matrices in the following way (see the above description of Rédei matrices):
a) a_{45} = 0, a_{11 }= 1, a_{21 }= 1, a_{31 }= 0 b) a_{45} = 0, a_{11 }= 1, a_{21 }= 0, a_{31 }= 1 c) a_{45} = 1, a_{11 }= 1, a_{21 }= 1, a_{31 }= 0 d) a_{45} = 1, a_{11 }= 0, a_{21 }= 1, a_{31 }= 1
We can describe the generic Rédei matrix R_{k} for the D_{2} family as follows, where the * entries would be entered in accordance with the above stipulations for matrices a, b, c, and d (cf. [20]):
[* 1 1 0 0
* 1 1 1 1
* 0 0 1 1
1 1 1 * *
1 1 1 * *]
In our examples that followwe utilize Lemmas 10 and 11 to demonstrate the existence of some new fields k as above that have infinite 2-class field tower and satisfy cases a, b, or c above, and we use [4] to obtain the 2-class numbers of our corresponding real quadratic number fields M.
Example 1: k = Q(√-5.13.7.827) = Q(√-376285), M = Q(√5.13.7.827) = Q(√376285), h(M) = 16, (5.13.7.827/2) = -1, (5/2) = (13/2) = (5/7) = (5/827) = (13/7) = (13/827) = -1, C_{k} has 4-rank 2 and the three nontrivial d_{k}-splittings of the second kind of k are (5.13, -4.7.827), (-5.827, 4.7.13), and (-13.827, 4.5.7). Since from Lemma 10 we see that k is a new field with infinite 2-class field tower and this cannot be obtained by using either Lemma 4 or Lemma 7, we can conclude that k is a new field of this type (see Case 60 in [20]).
Example 2: k = Q√-5.13.47.827) = Q(√-2526485), M = Q√5.13.47.827) = Q(√2526485), h(M) = 16, (5.13.47.827/2) = -1, (5/2) = (13/2) = (5/47) = (5/827) = (13/47) = (13/827) = -1, C_{k} has 4-rank 2. Since (2/7) = (2/47) = (827/7) = (827/47) = 1, we see from the above Kronecker symbol equalities that the primes 7 and 47 have the same Kronecker symbol formulations with the other primes dividing d_{k}, and from Lemma 10 we obtain a new field with infinite 2-class field tower.
Example 3: k = Q(√(-5.29.47.827) = Q(√-5636005), M = Q(√5.29.47.827) = Q(√5636005), h(M) = 16; C_{k} has 4-rank 2. Since (29/5) = 1 we see immediately from the proof of Lemma 11 that (5.29.47.827/2) = -1, and from Lemma 10 or Lemma 11 we are able to conclude that k is a new field with infinite 2-class field tower.
Example 4: Let k = Q(√-5.29.47.43) = Q(√-293045), and M = Q(√5.29.47.43) = Q(√293045). We see that h(M) = 16, (5.29.47.43/2) = -1, and in a similar manner to Examples 1, 2, and 3 we are able to conclude from Lemma 10 that k has infinite 2-class field tower.
Remark 8: Although our above examples illustrate that we can find new fields that satisfy the Rédei matrices listed in cases a, b, and c above, we are not able to utilize Lemma 10 to find a new field with infinite 2-class field tower that satisfies the Rédei matrix listed in case d. This can be seen from the fact that (p_{1}p_{2}q_{2}q_{3}/2) = 1, and from the fact that h(Q(√p_{1}p_{2}q_{2})) = h(Q(√p_{1}p_{2}q_{3})) = 4 (cf. Lemma 10).
We now put together Lemmas 10 and 11 and our above examples to state our first main result as the following theorem.
Theorem 1: Let F_{k} be the family of imaginary quadratic number fields k such that rank C_{k,2 }= 4, C_{k} has 4-rank 2, d_{k} is congruent to 4 mod 8, three negative prime discriminants divide d_{k}, and the Kronecker symbols of the primes dividing d_{k} have the format (p_{1}/q_{1}) = (p_{1}/q_{2}) = (p_{1}/q_{3}) = (p_{2}/q_{1}) = (p_{2}/q_{2}) = (p_{2}/q_{3}) = -1, where p_{1} and p_{2} are distinct primes dividing positive prime discriminants dividing d_{k}, and q_{1}, q_{2}, and q_{3} are distinct primes dividing negative prime discriminants dividing d_{k}.Then there exist new fields k in this family that have infinite 2-class field tower.
For the case when rank C_{k,2 }= 4 and 4-rank of C_{k} equal to 1, we know from Sueyoshi
(cf. Lemma 7: Part iii and [19]) that if five negative prime discriminants divide d_{k} and d_{k} is not congruent to 4 mod 8 then k has infinite 2-class field tower. For each of the remaining cases, i.e. when d_{k} is congruent to 4 mod 8 and exactly three or five negative prime discriminants divide d_{k}, and when exactly three discriminants divide d_{k} and d_{k} is not congruent to 4 mod 8,
we will show that there exists at least one new imaginary quadratic number field k with
infinite 2-class field tower. Furthermore, we will show that there are infinitely many new such fields k with infinite 2-class field tower in the case when exactly three discriminants divide d_{k} and d_{k} is not congruent to 4 mod 8. We begin with the case when five negative prime discriminants divide d_{k} and d_{k} is congruent to 4 mod 8.
Case 2: Five Negative Prime Discriminants Dividing d_{k}
with 4-Rank of C_{k} Equal to 1 and d_{k} Congruent to 4 mod 8
For the case when five negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, and d_{k} is congruent to 4 mod 8, we see from [19] that there are exactly six particular open Rédei sub-type matrices of 4-rank 1, and that k always has infinite 2-class field tower for two of these Rédei sub-type matrices (corresponding to matrices m and o in [19]). For the remaining two generic Rédei sub-type matrices (corresponding to matrices n and p in [19]) we now make use of Lemma 11 to show that there exists a new field k with infinite 2-class field tower that satisfies matrix n (resp. matrix p). We describe matrices n and p as follows, where once again it is understood that p_{1}* = -4, the matrix entry * signifies a choice of 0 or 1, and that these Rédei matrices are generic in the sense that the order of the q_{i}’s (2 ≤ i ≤ 5) can be changed. We divide each of the matrices n and p into its two possible generic Rédei matrices: n_{1, }n_{2}, p_{1}, p_{2} as follows:
n_{1}: [1 1 1 1 1 n_{2}: [0 1 1 1 1 p_{1}: [* 1 1 1 1 p_{2}: [* 1 1 1 1
1 1 1 1 1 0 1 1 1 1 * 0 1 1 0 * 0 1 1 0
* 0 1 1 0 * 0 1 1 0 1 0 0 1 1 0 0 0 1 1
* 0 0 1 1 * 0 0 1 1 1 0 0 1 1 0 0 0 1 1
* 0 1 0 1] * 0 1 0 1] * 1 0 0 1] * 1 0 0 1]
From the table on pages 337-338 in [19] we see that for matrix p_{2} it is known that k has infinite 2-class field tower, and therefore we eliminate matrix p_{2} in our subsequent formulations; furthermore we see that matrices p_{1, }n_{1}, and n_{2} are open Rédei matrices.
In order to make use of Lemma 10 to find new fields k that satisfy Rédei matrices n_{1}, n_{2}, and p_{1}, our next lemma demonstrates that we must have d_{M} congruent to 4 mod 8 for the corresponding real quadratic number field M given in Lemma 10.
Lemma 13: Let k satisfy Rédei matrices n_{1}, n_{2}, or p_{1}, and let M be any of the corresponding real quadratic number fields given in Lemma 10. If d_{M} is not congruent to 4 mod 8 then C_{M,2} is isomorphic to the group Z/2Z x Z/2Z.
Proof: If d_{M} is not congruent to 4 mod 8, then d_{M} = q_{2}.q_{3}.q_{4}.q_{5} for distinct primes q_{i} congruent to 3 mod 4, 2 ≤ i ≤ 5. For Rédei matrices n_{1} and n_{2} we have the following Rédei matrix:
R_{M} = [0 1 1 1
0 0 1 0
0 0 0 1
0 1 0 0]
Since R_{M} has rank 3 we see from the above Rédei matrix 4-rank property that C_{M} has 4-rank 0, and therefore by genus theory C_{M,2} is isomorphic to Z/2Z x Z/2Z. For Rédei matrix p_{1} we have the following Rédei matrix:
R_{M} = [1 1 1 1
0 1 1 1
0 0 0 1
1 0 0 0]
Once again we see that R_{M} has rank 3 and consequently C_{M,2} is isomorphic to Z/2Z x Z/2Z, which establishes our lemma.
We now show that when d_{M} is congruent to 4 mod 8 there exists a new field k that satisfies Rédei matrix n_{1} (resp. n_{2}, p_{1}).
Theorem 2: There exist new imaginary quadratic number fields k such that rank C_{k,2} = 4, C_{k} has 4-rank 1, five negative prime discriminants divide d_{k}, and d_{k} is congruent to 4 mod 8. In particular, for each possible open Rédei sub-type matrix of this family: n_{1}, n_{2}, p_{1}, there exists a new field k that satisfies the given open Rédei sub-type matrix.
Proof: We show that for each of n_{1}, n_{2}, p_{1}, there exists a new field k that satisfies the conditions of the theorem and that satisfies the given open Rédei sub-type matrix, which will prove our result. To establish our result we utilize Lemma 10, which necessitates finding a corresponding real quadratic number fields M to k such that h(M) ≥ 16 and for which the Kronecker symbol condition given in Lemma 10 is satisfied. From Lemma 13 we know that d_{M} must be congruent to 4 mod 8. The following fields respectively satisfy each of n_{1}, n_{2}, and p_{1}, and from [4] we see that they also satisfy the 2-class number bound condition of Lemma 10, with h(M) = 16, where k = Q(√-q_{2}q_{3}q_{4}q_{5}) and (without loss of generality) M = Q(√q_{2}q_{3}q_{4}).
for p_{1}: M = Q(√23.19.67) = Q(√29279)
for n_{1}: M = Q(√11.7.167) = Q(√12859)
for n_{2}: M = Q(√23.11.19) = Q(√4807)
By choosing our remaining prime q_{5} appropriately, which we do in Examples 5, 6, and 7 below, we are also able to obtain the second condition of Lemma 10, namely that (q_{2}q_{3}q_{4}/q_{5}) = -1, and therefore we are able to formulate new fields k = Q(√-q_{2}q_{3}q_{4}q_{5}) that respectively satisfy each of n_{1}, n_{2}, and p_{1}, and this proves our theorem.
As described in the proof of Theorem 2, for each of the following fields k = Q(√-q_{2}q_{3}q_{4}q_{5}) we have h(M) = 16 and (q_{2}q_{3}q_{4}/q_{5}) = -1, where M = Q(√q_{2}q_{3}q_{4}), and consequently the conditions of Lemma 10 are satisfied and we can conclude that k is a new field with infinite 2-class field tower.
Example 5: for p_{1}: k = Q(√-23.19.67.3) = Q(√-87837); q_{5} = 3, (-23/3) = 1, (-19/3) = (-67/3) = -1, (23.19.67/3) = -1, h(Q(√23.19.67)) = 16
Example 6: for n_{1}: k = Q(√-11.7.167.79) = Q(√-1015861); q_{5} = 79, (-17/79) = (-167/79) = -1, (-7/79) = 1, (11.7.167/79) = -1, h(Q(√11.7.167)) = 16
Example 7: for n_{2}: k = Q(√-23.11.19.103) = Q(√-495121); q_{5} = 103, (-23/103) = (-19/103) = -1, (-11/103) = 1, (23.11.19/103) = -1, h(Q(√23.11.19)) = 16
Case 3: Exactly Three Negative Prime Discriminants Dividing d_{k}
with 4-Rank of C_{k} Equal to 1 and d_{k} Congruent to 4 mod 8
For the case when exactly three negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, and d_{k} is congruent to 4 mod 8, we see from [20] that there are 34 open Rédei sub-type matrices of 4-rank 1, which are listed as belonging to Cases 56, 57, 58, 59, and 60 in [20]. We also see from [20] that all fields k with the Kronecker symbol format we described above in the 4-rank 2 case as designating Family D_{2} fields, i.e. (p_{1}/q_{2}) = (p_{1}/q_{3}) = (p_{2}/q_{2}) = (p_{2}/q_{3}) = (p_{1}/2) = (p_{2}/2) = -1 where p_{1} and p_{2} are primes congruent to 1 mod 4 dividing d_{k}, and q_{2} and q_{3} are primes congruent to 3 mod 4 dividing d_{k}, satisfy a Rédei sub-type matrix in Case 60 of [20]. We now designate fields of this type for the 4-rank 1 case as Family D_{1} fields, and fields for the 4-rank 1 case that satisfy an open Rédei sub-type matrix in Case 60 as Family D fields. We begin by establishing the following lemma, where the primes p_{1, }p_{2, }q_{2, }q_{3} are as above, and once again the negative prime discriminant q_{1}* = -4.
Lemma 14: Assume that exactly three negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, d_{k} is congruent to 4 mod 8, and that k satisfies an open Rédei sub-type matrix. Let M_{1,} M_{2}, and M_{3} denote the three corresponding real quadratic number fields given in Lemma 10 as follows (without loss of generality): M_{1} = Q(√q_{2}p_{1}p_{2}), M_{2} = Q(√q_{3}p_{1}p_{2}), M_{3} = Q(√p_{1}p_{2}q_{2}q_{3}). Then h(M_{1}) = h(M_{2}) = 4; and h(M_{3}) ≥ 8 if and only if k is a Family D field.
Proof: We give an illustration of the method by initially describing the five open Rédei sub-type matrices of 4-rank 1 belonging to Case 57 in [20] as follows, where a_{54} = 1, a_{21} = a_{51: } a) a_{41} = 0, a_{51} = 1, a_{31} = 1 b) a_{41} = 0, a_{51} = 1, a_{31} = 0 c) a_{41} = 1, a_{51} = 0, a_{31} = 1 d) a_{41} = 1, a_{51} = 0, a_{31} = 1 e) a_{41} = 1, a_{51} = 1, a_{31} = 1
We describe the generic Rédei matrix R_{k} for this family below, where the * entries would be entered in accordance with the above stipulations for matrices a, b, c, d, and e (cf. [20]):
[* 1 1 0 0
* 0 1 1 0
* 0 1 0 1
* 1 0 0 1
* 0 1 1 0]
It follows that the Rédei matrices of M_{1, }M_{2}, and M_{3 }for each of the five members of this family all have rank 3, and therefore in accordance with genus theory and the Rédei matrix 4-property we are able conclude that the 2-class groups of M_{1, }M_{2}, and M_{3} are isomorphic to Z/2Z x Z/2Z for each member of this family. We illustrate this result by displaying M_{1, }M_{2}, and M_{3} for the Rédei matrix of Case 57a, and we leave it to the reader to verify this for the remaining cases.
M_{1} = [0 1 0 0 M_{2} = [0 1 0 0 M_{3} = [1 1 1 0
1 0 1 0 1 0 0 0 0 0 0 1
0 1 0 1 0 0 1 1 1 0 0 1
1 0 1 1] 1 1 1 0] 0 1 1 1]
In a similar manner it can be readily shown that the same result applies to Cases 56, 58, and 59 as we have indicated applies to Case 57. However, for Case 60 with k as in the lemma, we demonstrate that we always obtain h(M_{1}) = h(M_{2}) = 4, h(M_{3}) ≥ 8.
We begin by designating the ten open Rédei sub-type matrices of 4-rank 1 belonging to Case 60 in [20] as follows, where a_{23} = a_{24} = a_{25} =a_{34 }= a_{35} = 1:
a) a_{45} = 0, a_{41} = a_{51} = 1, a_{11} = 0, a_{21} = a_{31} = 1
b) a_{45} = 0, a_{41} = a_{51} = 1, a_{11} = 0, a_{21} = 0, a_{31} = 0
c) a_{45} = 0, a_{41} = 1, a_{51} = 0, a_{11} = 0, a_{21} = 1, a_{31} = 0 d) a_{45} = 0, a_{41} = 1, a_{51} = 0, a_{11} = 0, a_{21} = 0, a_{31} = 1 e) a_{45} = 0, a_{41} = 1, a_{51} = 0, a_{11} = a_{21} = a_{31} = 1 f) a_{45} = 0, a_{41} = 1, a_{51} = 0, a_{11} = 1, a_{21} = a_{31} = 0 g) a_{45} = a_{41} = a_{51} = 1, a_{11} = 1, a_{21} = 0, a_{31} = 1 h) a_{45} = a_{41} = a_{51} = 1, a_{11} = a_{21} = a_{31} = 0 i) a_{45} = 1, a_{41} = 0, a_{51} = 1, a_{11} = 1, a_{21} = a_{31} = 0 j) a_{45} = 1, a_{41} = 0, a_{51} = 1, a_{11} = a_{21} = 0, a_{31} = 1
We describe the generic Rédei matrix R_{k} for this family below, where the * entries would be entered in accordance with the above stipulations for matrices a through j (cf. [20]):
[* 1 1 0 0
* 0 1 1 1
* 0 1 1 1
* 1 1 * *
* 1 1 * *]
It readily follows that for all the above fields we obtain h(M_{1}) = h(M_{2}) = 4, h(M_{3}) ≥ 8, and we illustrate this result by displaying M_{1, }M_{2}, and M_{3} for the Rédei matrix of Case 60g.
M_{1} = [0 1 0 0 M_{2} = [1 1 0 0 M_{3} = [0 1 1 1
0 1 1 1 1 1 1 1 0 1 1 1
1 1 0 1 1 1 0 1 1 1 1 1
1 1 1 0] 1 1 1 0] 1 1 1 1]
Since M_{1} and M_{2} have ranks 3 and M_{3} has rank 2, we see from genus theory and the Rédei matrix 4-property that the 2-class groups of M_{1} andM_{2} are isomorphic to Z/2Z x Z/2Z, and that M_{3} has 4-rank 1 and consequently has order greater than or equal to 8. In a similar way it can readily be shown that this same result applies to all members of the above family (we once again leave the remaining cases for the reader to verify), which establishes our lemma.
We now demonstrate the existence of a new field in the case when exactly three negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, and d_{k} is congruent to 4 mod 8.
Theorem 3: There exists a new field k such that exactly three negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, and d_{k} is congruent to 4 mod 8.
Proof: We first note that we can only apply Lemma 10 to fields that satisfy Rédei sub-type matrices 60e, f, g, i, since we have (q_{2}q_{3}q_{4}q_{5}/2) = -1 for these matrices and (q_{2}q_{3}q_{4}q_{5}/2) = 1 for the other six Rédei sub-type matrices listed above. We establish our theorem by applying Lemmas 10 and 14 to the field k = Q(√-5.13.7.83) = Q(√-37765), which satisfies Rédei matrix 60g. In accordance with Lemma 14 we see that M_{3} = Q(√5.13.7.83) = Q(√37765) and that h(M_{3}) ≥ 8. From [4] we see that h(M_{3}) = 16, and since 5 and 13 are congruent to 5 mod 8, 7 is congruent to 7 mod 8, and 83 is congruent to 3 mod 8, we obtain that (5.13.7.83/2) = -1, and we observe that the conditions of Lemma 11 are satisfied. Consequently from Lemma 10 we can conclude that k has infinite 2-class field tower, and since k satisfies Rédei matrix 60g we see that k is a new field, which proves our theorem.
Remark 9: We expect that using the technique described in Theorem 3, one can readily find additional new fields k as above, and we leave this to the interested reader to explore. However, when d_{k }is congruent to 4 mod 8 and C_{k} has 4-rank equal to 1, the 4-rank of C_{j} for a corresponding field j obtained by using CRT and DPAP as in Lemma 12 may not be the same as the 4-rank of C_{k}. To see an example of this, let k = Q(√-3.5.7.29) = Q(√-3045) and j = Q(√-3.263.5.7.29) = Q(√-800835). We see from the above Rédei matrix 4-property that C_{k }has 4-rank 1 and C_{j} has 4-rank 0, and we also see that the field j is obtained from the field k by CRT and DPAP as described in Lemma 12. In addition, we see from [20] that field k belongs to Case 59, field j belongs to Case 34 (see below) and it is not known if either of these fields have infinite 2-class field tower. However, if d_{k} is not congruent to 4 mod 8 then for all fields in the infinite collection of fields described in Lemma 12, the 4-ranks of C_{k} and C_{j} will be the same.
We next show that in the remaining case when the 4-rank of C_{k} is equal to 1, exactly three negative prime discriminants divide d_{k}, and d_{k} is not congruent to 4 mod 8, there are infinitely many new fields k that satisfy the 2-class field tower conjecture.
Case 4: Exactly Three Negative Prime Discriminants Dividing d_{k}
with 4-Rank of C_{k} Equal to 1 and d_{k} Not Congruent to 4 mod 8
For the case when exactly three negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, and d_{k} is not congruent to 4 mod 8, we see from [20] that there are seven open Rédei sub-type matrices of 4-rank 1, which are listed as matrices 31, 33, 35, 43, 47, 48, and 49 in [20], for which we describe as follows, using the above description of Rédei matrices.
Matrix 31: a_{12 }=a_{13} = a_{15} = a_{23} = a_{24} = a_{25} = a_{34} = a_{45} = 1, a_{14} = a_{35 }= 0
Matrix 33: a_{12 }=a_{13} = a_{15} = a_{23} = a_{24} = a_{25} = a_{34} = a_{35} = a_{45} = 1, a_{14} = 0
Matrix 35: a_{12 }=a_{13} = a_{14} = a_{15} = a_{23} = a_{24} = a_{25} = a_{35} = a_{45} = 1, a_{34} = 0
Matrix 43: a_{12} = a_{14} = a_{23} = a_{24} = a_{35} = a_{45} = 1, a_{13} = a_{15} = a_{25} = a_{34} = 0
Matrix 47: a_{12} = a_{15} = a_{23} = a_{25} = a_{34} = a_{35} = a_{45} = 1, a_{13} = a_{14} = a_{24} = 0
Matrix 48: a_{12} = a_{15} = a_{23} = a_{24} = a_{34} = a_{35} = a_{45} = 1, a_{13} = a_{14} = a_{25} = 0
Matrix 49: a_{12} = a_{15} = a_{23} = a_{24} = a_{25} = a_{34} = a_{35} = a_{45} = 1, a_{13} = a_{14} = 0
We now make use of this categorization to establish the following theorem:
Theorem 4: There are infinitely many new fields k such that exactly three negative prime discriminants divide d_{k}, the 4-rank of C_{k} is equal to 1, and d_{k} is not congruent to 4 mod 8.
Proof: Let k = Q(√-q_{1}q_{2}q_{3}p_{1}p_{2}) where q_{1, }q_{2}, and q_{3} are distinct negative prime discriminants, p_{1 }andp_{2} are distinct positive prime discriminants, and d_{k} is not congruent to 4 mod 8. We determine which of the above seven open Rédei sub-type matrices of 4-rank 1 are candidates for new fields that satisfy a Rédei sub-type matrix. From the Rédei matrix 4-rank property and genus theory we see that C_{M,2} is isomorphic to Z/2Z x Z/2Z for Rédei sub-type matrices 31, 43, 47, and 48, for all corresponding real quadratic number fields M to k in Theorem 11, since the ranks of these Rédei matrices are all 3. Therefore we are not able to use Theorem 11 to obtain new fields that satisfy these Rédei sub-type matrices. For Rédei sub-type matrix 49, we see from the Rédei matrix 4-rank property and genus theory that C_{M,2} is isomorphic to Z/2Z x Z/2Z for M = Q(√q_{1}q_{2}p_{1}p_{2}) and M = Q(√q_{1}q_{3}p_{1}p_{2}), and that C_{M} has 4-rank 1 for M = Q(√q_{2}q_{3}p_{1}p_{2}). However, (q_{2}q_{3}p_{1}p_{2}/q_{1}) = 1 and therefore the Kronecker symbol requirement of Lemma 10 is not satisfied for fields that satisfy Rédei sub-type matrix 49. For Rédei sub-type matrices 33 and 35, although we obtain using the above methods that C_{M,2} is isomorphic to Z/2Z x Z/2Z for two of the three possible fields M in each type, for Rédei sub-type matrix 33 with M = Q(√q_{2}q_{3}p_{1}p_{2}) we obtain that C_{M} has 4-rank 1 and (q_{2}q_{3}p_{1}p_{2}/q_{1}) = -1, and for Rédei sub-type matrix 35 with M = Q(√q_{1}q_{2}p_{1}p_{2}) we obtain that C_{M} has 4-rank 1 and (q_{1}q_{2}p_{1}p_{2}/q_{3}) = -1. We illustrate this result for M = Q(√q_{2}q_{3}p_{1}p_{2} in matrix 33, and we leave the remaining details for the reader to check.
Given the above Rédei matrix entries for matrix 33 we see that the Rédei matrix for
M = Q(√q_{1}q_{2}p_{1}p_{2}) is the following:
R_{M} = [0 1 1 1
0 1 1 1
1 1 1 1
1 1 1 1]
It therefore follows that R_{M} has rank 2 and by the Rédei matrix 4-rank property we are able to conclude that C_{M} has 4-rank 1. From the listing of the Rédei matrix entries for matrix 33 we seethat (q_{2}q_{3}p_{1}p_{2}/q_{1}) = -1.
We thus see that Rédei sub-type matrices 33 and 35 are the only possible candidates for new fields that satisfy a Rédei sub-type matrix of 4-rank 1 in the case that we are now considering.
From [11] we see that for the field M= Q(√19.11.13.41) = Q(√111397) we have h(M) = 16, and we are able to use this field M to obtain an imaginary quadratic number field k that satisfies Rédei sub-type matrix 33 and the conditions of Lemma 10 (see Example 8 and Remark 10 below). Consequently from Lemmas 10 and 12, and Remark 8, we are able to conclude that there are infinitely many new fields that satisfy the conditions of our theorem.
Example 8: Let k = Q(√-19.11.191.13.41) = Q(√-21276827); q_{3} = 191, (19/191) = (11/191) = (41/191) = -1, (13/191) = 1, (19.11.13.41/191) = -1, h(Q(√19.11.13.41)) = 16; we see that
k satisfies Rédei sub-type matrix 33, and consequently from Lemma 10 we obtain that k is a
new field with infinite 2-class field tower, and from Lemma 12 and Remark 8 we conclude that there are infinitely many new fields with infinite 2-class field tower that satisfies the conditions of Theorem 2.
Remark 10: We expect that by the use of [4] and Lemmas 10 and 12, one can readily obtain infinitely many new fields that satisfy Rédei sub-type matrix 35, and we leave this task to the interested reader. We note that if M is a real quadratic number field corresponding to a field k as in Lemma 11, such that h(M) ≥ 16 and the Kronecker symbols of the primes dividing d_{M} are consistent with the entries of Rédei sub-type matrix 35 (resp. matrix 33), then by CRT and DPAP (see the proof of Lemma 13) we know that there exist infinitely many fields j with infinite 2-class field tower that satisfy Rédei sub-type matrix 35 (resp. 33), and since d_{k} is not congruent to 4 mod 8 we see from Remark 8 that the 4-rank of C_{j} is equal to 1 for all fields j in this infinite collection and consequently we would obtain infinitely many new fields.
We summarize our results thus far for obtaining new fields k that satisfy the 2-class field tower conjecture in the following table, where once again k is an imaginary quadratic number fields such that rank C_{k,2} = 4, “Negative Prime Discriminants” denotes the exact number of negative prime discriminants dividing d_{k}, “Congruency” denotes whether or not d_{k} is congruent to 4 mod 8, “Examples” denotes the examples we have supplied, and q_{3} denotes a prime congruent to 3 mod 4 that has the same kronecker symbols with the primes 19, 11, 13, and 41 as does the prime 191, as described in the proof of Lemma 12.
Table 2: New Fields that Satisfy the 2-Class Field Tower Conjecture
4-rank of C_{k} Negative Prime Discrminants Congruency Examples
2 3 4 mod 8 k = Q(√-5.13.7.827)
k = Q(√-5.13.47.827)
k = Q(√-5.29.47.827)
k = Q(√-5.29.47.43)
1 5 4 mod 8 k = Q(√-23.19.67.3)
k = Q(√-11.7.167.79)
k = Q(√-23.11.19.103)
1 3 4 mod 8 k = Q(√-5.13.7.83)
1 3 not 4 mod 8 k = Q(√-19.11.191.13.41)
Infinitely Many Fields
of the form
k = Q(√-19.11.q_{3}.13.41)
We next examine the case when C_{k} has 4-rank 0 instead of 4-rank 1 (i.e. C_{k,2} is elementary), and exactly three or five negative prime discriminants divide d_{k} for the fields k that we are considering (i.e. once again k imaginary quadratic with rank C_{k,2} = 4), to determine if there are new fields k that satisfy the conditions of Lemma 10. We begin with the five negative prime discriminant case.
Case 5: Five Negative Prime Discriminants Dividing d_{k}
with 4-Rank of C_{k} Equal to 0
To begin with, we demonstrate that there are no new fields k that satisfy the conditions of Lemma 10 for the five negative prime discriminant case when the 4-rank of C_{k} is equal to 0. We first obtain from [19] that there are two open Rédei sub-type matrices such that d_{k} is not congruent to 4 mod 8 (Rédei matrices k and l in [19]), which are given in [19] as follows
(we designate these matrices as “A” and “B” for ease of notation):
A = [1 1 1 0 1 B = [0 1 1 0 0
0 0 1 1 0 0 0 1 1 0
0 0 0 1 1 0 0 0 1 1
1 0 0 0 1 1 0 0 0 1
0 1 0 0 1] 1 1 0 0 0]
The following lemma shows that if k satisfies Rédei matrix A or B then the required condition in Lemma 11 that h(M)| ≥ 16 is not satisfied.
Lemma 15: Let k satisfy Rédei matrices A or B (as described above), and let M be any of the corresponding real quadratic number fields given in Lemma 11. Then C_{M,2} is isomorphic to Z/2Z x Z/2Z.
Proof: Let k = Q(√-q_{1}q_{2}q_{3}q_{4}q_{5}) and M = Q(√q_{i}q_{j}q_{k}q_{l}), q_{i} congruent to 3 mod 4, 1 ≤ i ≤ 5, q_{i} all distinct and {i, j, k, l} C {1, 2, 3, 4, 5}. If k satisfies Matrix A then we have the following Kronecker symbols: (-q_{1}/q_{2}) = (-q_{1}/q_{3}) = (-q_{2}/q_{3}) = (-q_{2}/q_{4}) = (-q_{3}/q_{4}) = (-q_{3}/q_{5}) = (-q_{1}/q_{5}) = (-q_{4}/q_{5}) = -1, (-q_{1}/q_{4}) = (-q_{2}/q_{5}) = 1. Consequently we have the following possible Rédei matrices for M:
M_{1} = Q(√q_{1}q_{2}q_{3}q_{4}) M_{2} = Q(√q_{1}q_{2}q_{3}q_{5}) M_{3} = Q(√q_{1}q_{2}q_{4}q_{5}) M_{4} = Q(√q_{3}q_{2}q_{4}q_{5})
[1 1 1 0 [0 1 1 1 [0 1 0 1 [1 1 0 1
0 0 1 1 0 0 1 0 0 0 1 0 0 1 1 1
0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 1
1 1 0 0] 0 1 0 0] 0 1 0 0] 0 0 0 1]
M_{5} = Q(√q_{2}q_{3}q_{4}q_{5})
[1 1 1 0
0 1 1 1
0 0 0 1
1 0 0 0]
Since all matrices M_{1},M_{2}, M_{3}, M_{4}, M_{5} have rank 3, we see from genus theory and the Rédei matrix 4-rank property that the 4-rank of M is equal to 0 and thus C_{M,2} is isomorphic to Z/2Z x Z/2Z. In a similar way, it can be shown that if k satisfies Matrix B then C_{M,2} is isomorphic to Z/2Z x Z/2Z (we once again leave the details to the reader) and consequently our lemma is proved.
For the case when five negative prime discriminants divide d_{k}, C_{k,2} is elementary, and d_{k} is congruent to 4 mod 8, we see from [19] that there are three open Rédei sub-type matrices which we designate as matrices C, D_{1}, and D_{2} (derived from matrices o and p in [19]), which we give as follows, where once again the * signifies a choice of 0 or 1:
C = [* 1 1 1 1 D_{1} = [* 1 1 1 1 D_{2} = [* 1 1 1 1
* 0 1 0 1 * 0 1 1 0 * 0 1 1 0
* 0 0 1 1 1 0 0 1 1 0 0 0 1 1
* 1 0 0 1 1 0 0 1 1 0 0 0 1 1
1 0 0 0 0] * 1 0 0 1] * 1 0 0 1]
The following lemma shows that if k satisfies Rédei matrices C, D_{1}, or D_{2}, then k does not satisfy the conditions of Lemma 10.
Lemma 16: Let k = Q(√-q_{2}q_{3}q_{4}q_{5}) with q_{i} congruent to 3 mod 4, 2 ≤ i ≤ 5, q_{i} all distinct, let k satisfy Rédei matrices C, D_{1}, or D_{2} (as described above), and let M be any of the corresponding real quadratic number field given in Lemma 5. Then either C_{M,2} is isomorphic to Z/2Z x Z/2Z, or M =Q(√q_{j}q_{j}q_{k}) with i, j, k distinct, {i, j, k} C {2, 3, 4, 5} = {i, j, k, l}, and (q_{j}q_{j}q_{k}/q_{l}) = 1.
Proof: If k satisfies Matrix C then we have the following Kronecker symbols: (-q_{2}/q_{3}) = (-q_{3}/q_{4}) = (-q_{3}/q_{5}) = (-q_{4}/q_{5}) = (-q_{2}/q_{4}) = (-q_{2}/q_{5}) = -1, q_{5} congruent to 7 mod 8. If M = Q(√q_{2}q_{3}q_{4}q_{5}) then we see that M has rank 3 and therefore the 4-rank of C_{M }is equal to 0 and C_{M,2} is isomorphic to Z/2Z x Z/2Z . If M is not equal to Q(√q_{2}q_{3}q_{4}q_{5}) then M has the format given in the lemma, and we see that (q_{2}q_{3}q_{4}/q_{5}) = (q_{2}q_{3}q_{5}/q_{4}) = (q_{2}q_{4}q_{5}/q_{3}) = (q_{3}q_{4}q_{5}/q_{2}) = 1, which establishes our lemma when k satisfies Matrix C. If k satisfies Matrices D_{1 }or D_{2} then we have the following Kronecker symbols: (-q_{2}/q_{3}) = (-q_{3}/q_{4}) = (-q_{3}/q_{5}) = (-q_{4}/q_{5}) = (-q_{2}/q_{4}) = -1, (-q_{2}/q_{5}) = 1. If M = Q(√q_{2}q_{3}q_{4}q_{5}), Q(√q_{2}q_{3}q_{4}), Q(√q_{2}q_{3}q_{5}), or Q(√q_{3}q_{4}q_{5}), then once again we see that M has rank 3 and thus the 4-rank of C_{M} is equal to 0 and C_{M,2} is isomorphic to Z/2Z x Z/2Z . If M = Q(√q_{2}q_{4}q_{5}) then (q_{2}q_{4}q_{5}/q_{3}) = 1, and if M = Q(√q_{3}q_{4}q_{5}) then (q_{3}q_{4}q_{5}/q_{2}) = 1; consequently our lemma has been proved.
We therefore are able to conclude from Lemmas 15 and 16 that Lemma 10 is not applicable in the case when five negative prime discriminants divide d_{k} and C_{k,2} is elementary, and consequently by our present techniques we are not able to formulate any new fields that have infinite 2-class field tower in this case. We now examine the case when exactly three negative prime discriminants divide d_{k} and C_{k,2} is elementary.
Case 6: Exactly Three Negative Prime Discriminants Dividing d_{k}
with 4-Rank of C_{k} Equal to 0
We demonstrate that there are also no new fields k that satisfy the conditions of Lemma 10 when exactly three negative prime discriminants divide d_{k} and the 4-rank of C_{k} is equal to 0. We first obtain from [20] that there are seven open Rédei sub-type matrices such that d_{k} is not congruent to 4 mod 8 (Rédei matrices 16, 28, 30, 32, 34, 49 in [20]), which are given in [20] as follows, where we designate Cases 34a and 34b to represent the two possible open Rédei sub-type matrices for Case 34, where once again the * signifies a choice of 0 or 1:
#16 = [1 1 1 1 1 #28 = [1 1 1 0 1 #30 = [1 1 1 0 1
0 0 1 0 1 0 1 1 1 1 0 0 1 1 0
0 0 1 1 0 0 0 1 0 1 0 0 0 1 1
1 0 1 1 1 0 1 0 0 1 0 1 1 1 1
0 1 0 1 0] 1 1 1 1 0] 1 0 1 1 1]
#32 = [0 1 1 1 1 #34a = [0 1 1 1 1 #34b = [0 1 1 1 1
0 0 1 1 0 0 0 1 0 1 0 0 1 0 1
0 0 0 1 1 0 0 0 1 1 0 0 0 1 1
0 1 1 1 1 1 0 1 0 0 1 0 1 1 1
1 1 0 1 1] 1 1 1 0 1] 1 1 1 1 0]
#49 = [0 1 0 0 1
0 1 1 1 1
1 0 1 1 1
0 1 1 0 0
1 1 1 0 1]
The following lemma shows that if k satisfies any of the above seven Rédei matrices then k does not satisfy the conditions of Lemma 10.
Lemma 17: Let k = Q(√-q_{1}q_{2}q_{3}p_{1}p_{2}) where q_{1, }q_{2}, and q_{3} are distinct negative prime discriminants, p_{1 }andp_{2} are distinct positive prime discriminants, and d_{k} is not congruent to 4 mod 8. Let k satisfy Rédei matrices 16, 28, 30, 32, 34a, 34b, or 49, and let M be any of the
corresponding real quadratic number field given in Lemma 10. Then either C_{M,2} is isomorphic to
Z/2Z x Z/2Z, or (without loss of generality) M = Q(√q_{1}q_{2}p_{1}p_{2}) with (q_{1}q_{2}p_{1}p_{2}/q_{3}) = 1.
Proof: We illustrate the proof by examining matrices 16 and 34a. For matrix 16 we have the following three possible matrices for M:
M_{1} = [1 1 0 0 M_{2} = [1 1 1 0 M_{3} = [1 1 1 0
0 0 1 0 0 0 0 1 0 1 1 0
0 1 0 1 1 0 0 1 1 1 1 1
1 0 1 0] 0 1 1 0] 0 0 1 1]
Since the rank of each of the above three matrices is 3, we see from genus theory and the Rédei matrix 4-rank property that C_{M,2} is isomorphic to Z/2Z x Z/2Z for M as stated in the lemma. It can readily be determined that this same result holds for matrices 28, 30, and 32, and we once again leave the details for the reader to check. For matrix 34a we have the following three possible matrices for M, where M_{3} = Q(√q_{1}q_{3}p_{1}p_{2}):
M_{1} = [1 1 0 1 M_{2} = [0 1 1 1 M_{3} = [0 1 1 1
0 0 1 1 0 0 0 1 0 1 1 1
0 1 1 0 1 0 1 0 1 1 0 0
1 1 0 0] 1 1 0 0] 1 1 0 0]
Since matrices M_{1} and M_{2} have rank 3 and matrix M_{3} has rank 2, we see as above that C_{M,2} is isomorphic to Z/2Z x Z/2Z when M = M_{1} or M_{2}, and C_{M} has 4-rank 1 if M = M_{3}. However, since (q_{1}q_{3}p_{1}p_{2}/q_{2}) = 1 our lemma is established for matrix 34a. Similarly it can be shown that for matrices 34b and 49 we obtain that C_{M,2} is isomorphic to Z/2Z x Z/2Z when M = M_{1} or M_{2}, and C_{M} has 4-rank 1 if M = M_{3} with (q_{1}q_{3}p_{1}p_{2}/q_{2}) = 1 (without loss of generality; we once again leave the details to the reader to check), and consequently our lemma has been proved.
Finally, we examine the case when exactly three negative prime discriminants divide d_{k}, C_{k,2} is elementary, and d_{k} is congruent to 4 mod 8. We observe from [20] that there are 26 open Rédei sub-type matrices, which belong to Cases 56, 57, 58, and 59 in [20]. As an illustration we list the eight open Rédei sub-type matrices for Case 59 as follows, where a_{12} = a_{13} = a_{23} = a_{24} = a_{34} = a_{35} = 1, a_{14} = a_{15} = a_{25} = 0:
a) a_{21} = a_{31} = a_{41} = a_{51} = 1, a_{45} = 0
b) a_{21} = a_{31} = a_{51} = 1, a_{45} = a_{41} = 0
c) a_{41} = a_{51} = 1, a_{45} = a_{21} = a_{31} = 0
d) a_{45} = a_{41} = a_{51} = 1, a_{21} = a_{31} = 0
e) a_{45} = a_{31} = a_{41} = a_{51} = 1, a_{21} = 0
f) a_{45} = a_{21} = a_{31} = a_{41} = 1, a_{51} = 0
g) a_{45} = a_{51} = 1, a_{21} = a_{31} = a_{41} = 0
h) a_{45} = a_{31} = a_{51} = 1, a_{21} = a_{41} = 0
Using similar techniques to what we have described above, it can readily be shown that C_{M,2} is isomorphic to Z/2Z x Z/2Z for all corresponding real quadratic number fields M for each of the above 26 open Rédei sub-type matrices (we once again leave the details to the reader to verify) and consequently we see that there are no new fields k that satisfy the conditions of Lemma 10 when exactly three negative prime discriminants or five negative prime discriminants divide d_{k} and the 4-rank of C_{k} is equal to 0.
Open Questions on the 2-Class Field Tower Conjecture
Although we have found new fields k, where k is an imaginary quadratic number field with rank C_{k,2 }= 4 and C_{k} having 4-rank 1 or 2 that satisfy the 2-class field tower conjecture, and infinitely many new fields in the 4-rank 1 case, there are still many open questions. We state the following five open questions.
Question 1: Do there exist infinitely many new fields k as above such that rank C_{k,2 }= 4 and C_{k} has 4-rank 2?
Question 2: Do there exist new fields k such that rank C_{k,2 }= 4 and C_{k} has 4-rank 2 for case d as described in the 4-rank of C_{k} equal to 2 section? (see Question 4)
Question 3: Do there exist infinitely many new fields k such that rank C_{k,2 }= 4 and C_{k} has 4-rank 1 in the cases when d_{k} is congruent to 4 mod 8 and exactly three negative prime discrimants or five negative prime discriminants divide d_{k}?
Question 4: Do there exist new fields k as above that do not satisfy the conditions of Lemma 11, such that k has infinite 2-class field tower? Perhaps such a field k can be obtained by satisfying the conditions of Lemma 8 (cf. Prop. 3.4 in [14]), but we have not found any such fields k that satisfy the conditions of Lemma 8 without also satisfying the conditions of Lemmas 4, 5, or 7. We note that any new field satisfying the conditions given in Question 2 cannot be obtained from Lemma 10 (cf. Remark 8).
Question 5: Do there exist new imaginary quadratic number fields k with rank C_{k,2 }= 4 that satisfy the conditions of Lemma 10 and consequently have infinite 2-class field tower in the case when the 4-rank of C_{k} is 0, for which exactly three or five negative prime discriminants divide d_{k} and for which d_{k} is congruent to 4 mod 8 or not congruent to 4 mod 8?
Acknowledgements
I would like to thank Chip Snyder for a number of productive discussions concerning the results of this paper, and Yukata Sueyoshi for his helpful personal communication conveying to me that my field in Example 1 is indeed a new field, as well as for sending me his related 2010 paper.
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