Talk:Supersingular prime

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

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

message: 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)
replies: text: 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)

text: MathWorld has professional editors and is considered by Wikipedia editors to satisfy Wikipedia:Reliable sources. There is a template to use MathWorld as reference:  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)
thread_title: Reference needed

message: 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)
thread_title:  Complete rewrite 

message: 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  Fp. 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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
thread_title:  Equivalences