Talk:Supersingular prime

Untitled

Should this page be split into two pages -- one for each notion of supersingular? Minded17 04:04, 11 February 2007 (UTC)

A supersingular elliptic curve iover a field with p elements is one with Hasse invariant = 0; it is equivalent to have no non-trivial p-torsion, and there are some other conditions. The usage here is not exactly familiar to me; I'll have to look into it.

Charles Matthews 21:24, 15 Jun 2004 (UTC)

Formatting

Could someone clean this up with some LaTEX formatting? SigmaEpsilon → ΣΕ 04:59, 9 May 2006 (UTC)

Reference needed

I have never heard of a prime p such that X_0(p)^+ has genus zero called a supersingular prime (and I have written papers about modular curves). Moreover it seems like one does not need a special word to mean "either 2 or 3 or .... or 71." If a reference for this usage cannot be found, the second definition will be deleted. Plclark 09:38, 8 October 2007 (UTC)Plclark

The article already has a reference to http://mathworld.wolfram.com/SupersingularPrime.html which lists both definitions. MathWorld is considered a reliable source and the MathWorld article is referenced but I haven't seen the references. Our article's list of 15 supersingular primes also has a reference to http://www.research.att.com/~njas/sequences/A002267 in OEIS. PrimeHunter 00:01, 9 October 2007 (UTC)

I and other research mathematicians absolutely do not consider MathWorld to be a reliable source. If it is standard wikipedic practice to do so, please let me know: this seems to me like a big mistake.

I have looked at some of the sources mentioned in the MathWorld article. Going from the bottom up: Sloane's page uses the word supersingular, but refers back to the MathWorld page, so does not serve to establish the usage of the term. The two books on elliptic curves by Silverman do not discuss supersingular in the sense of genera of X_0(p)^+. I cannot access Ogg's paper on the internet. I did find the article by Conway and Norton, and it does not use the word "supersingular." I have also not been able to access The Atlas of finite groups, and by the way, it is unacceptable to reference an entire book without giving a page number. This contradicts the well-held idea that the burdensome aspects of verifiability should be assumed by the author, not the reader.

So let me ask: if I look through the remaining two sources and don't find supersingular used in this way, what will happen then?

Related to this issue is the following sentence: "It is also possible to define supersingular primes in a number-theoretic way using supersingular elliptic curves defined over the algebraic closure of the finite field GF(p) that have their j-invariant in GF(p^2)."

First, as written it is not sufficiently clear which sense of supersingular this refers to -- I gather that it it supposed to refer to the supersingular in the "modular" sense and to justify the name by relating it to supersingular in the arithmetic geometric sense. But the sentence does not do so and is not clear (even) to an expert reader: in fact every supersingular elliptic curve can be defined over GF(p^2), and this necessary condition is not, for sufficiently large p, sufficient. So what is actually intended?

As an expert, I should say that there is some relationship between supersingular elliptic curves and the genera of modular curves. For instance, the curves for which X_0(p) (but not X_0^+(p)) have genus zero are exactly those for which there exists exactly one supersingular elliptic curve modulo p. So there will be a similar, but more complicated, condition, expressing the fact that the genus of X_0^+(p) is zero in terms of supersingular j-invariants and fixed points of the Atkin-Lehner involution w_p. But again, this information either needs to appear in the article, or it needs to be given a precise reference. Plclark 22:06, 14 October 2007 (UTC)Plclark

MathWorld has professional editors and is considered by Wikipedia editors to satisfy Wikipedia:Reliable sources. There is a template to use MathWorld as reference: {{MathWorld}} with hundreds of uses. I know MathWorld contains errors; probably significantly more than peer reviewed journals. I have personally submitted several corrections to MathWorld articles about prime numbers. http://mathworld.wolfram.com/SupersingularPrime.html is authored by John McKay (mathematician) and http://www.research.att.com/~njas/sequences/A002267 by Neil Sloane personally with a low index (2267) which is sometimes considered a sign the sequence may be significant. http://mathworld.wolfram.com/MonsterGroup.html references Ogg's paper for a result about supersingular primes (but it doesn't whether the paper uses the term and I don't know). I think there is sufficient online evidence to include the alternative definition of supersingular prime. Maybe there isn't enough evidence to create an article if it was the only meaning of supersingular prime, but an alternative meaning in an existing article requires less. PrimeHunter 00:58, 15 October 2007 (UTC)

As I said, it may be the case that some wikipedia editors use MathWorld as a primary source, but since no one in academia does so, by perpetuating this practice you lessen the chance of wikipedia being a credible reference to students and researchers (which I assume is most of your intended audience for "supersingular prime").

I will check Ogg's paper to see whether he uses supersingular in the "modular" sense. If he does not, then there is no evidence that anyone except John McKay has ever used it in any context except for the MathWorld article. In this case, I think the entry should be revised to say that supersingular is used in this other context on MathWorld and on Neil Sloane's page but is otherwise not standard.

Your comments on the "low number" of the sequence is not relevant: there is no doubt that the sequence in question is an extremely famous and important one: it is the subject of Moonshine. What is at issue is whether we should promote the practice of calling this sequence of primes "supersingular", a term which is already used for something else and which has thus far zero support in primary sources.

Finally, you did not justify the sentence "It is also possible to define..." I will remove it for now. Of course if someone wants to fix it later, they can do so. Plclark 01:59, 15 October 2007 (UTC)Plclark

Complete rewrite

Recently I received an email from (the eminent mathematician) John McKay which clarified several points, both mathematical and historical. The second use of supersingular is not due to him but rather stems from work of Andrew Ogg. Moreover, he informed me of the precise characterization of such primes in terms of supersingular j-invariants of elliptic curves (a characterization whose existence I alluded to above). This characterization is sort of obliquely waved at in the Mathworld article; an earlier version of this wikipedia article had the merit of being less vague (but the drawback of being incorrect).

I also noticed that the definition of supersingular elliptic curve given in this article is incorrect: it said $\mathbb {F} _{p}$ whereas it should have said ${\overline {\mathbb {F} _{p}}}$.

His email contained the following sentence: "I agree with you strongly that it is unwise for Wikipedia to quote Mathworld for reference."

I was therefore motivated to completely rewrite the article and have just done so. It seemed appropriate to outsource the definition of supersingular elliptic curve to another article; note however that although this link exists, the discussion on that page is not yet satisfactory.

In this sitting I concentrated my efforts on giving the two definitions of supersingular prime and distiguinshing between them, with appropriate historical context. I therefore took out some very worthwhile (but too brief) remarks on the set of supersingular elliptic primes for an elliptic curve. To my mind, such results -- of Deuring, Serre, Lang-Trotter, Elkies, and others -- deserve a much more careful and complete treatment. If no one else does so, then one day I will take a crack at it. 72.152.92.55 (talk) 17:29, 26 November 2007 (UTC)Plclark

Equivalences

I don't see how the equivalences 1.-3. relate to the first notion of supersingular primes: They only relate the second notion to the notion of supersingular curves over arbitrary characteristic-p fields, while the first definition of supersingular primes involves supersingular curves over F_p. So, IMO, the two concepts are not related to each other, only both related to "supersingular elliptic curves".--Roentgenium111 (talk) 22:52, 11 May 2009 (UTC)

I'm not sure I understand your point. One piece of information (which you may already be aware of): any supersingular elliptic curve has j-invariant either in F_p or in F_{p^2}. But in general, yes -- the two concepts are related insofar as they both relate to supersingular elliptic curves. (I didn't choose any of this terminology, but all of it does indeed appear in the literature.) What specific change do you have in mind? Plclark (talk) 01:44, 12 May 2009 (UTC)

I would replace the last part of the sentence "Although these two usages are certainly distinct (the first is relative to a particular elliptic curve, whereas the second is not), they are related." by "..., they are both related to supersingular elliptic curves: For the first one this is in the definition, and for the second, we have the equivalence:....". Alternatively, I would consider splitting the article in two for the two distinct definitions.--Roentgenium111 (talk) 13:57, 13 May 2009 (UTC)

Both of these suggestions sound reasonable to me. In fact, the latter, bolder one seems like a clear improvement. Go for it! Plclark (talk) 21:31, 21 May 2009 (UTC)