From: victorsm@home.net (Victor S. Miller) Subject: Re: Possible Torsion on Elliptic Curves Date: Fri, 19 Feb 1999 14:28:15 GMT Newsgroups: sci.math,sci.math.research Keywords: Possible Torsion on Elliptic Curves over number fields Roberto Maria Avanzi writes: > I need to know if there is an explicit answer to this question > > Which possible groups appear as groups of rational torsion > points on Elliptic Curves over the field Q(\sqrt{-7}) ? The problem of boundedness of torsion for elliptic curves over quadratic fields was settled by Sheldon Kamienny in a series of papers. He showed that if K is ANY quadratic number field, if p is a prime dividing the order of the torsion subgroup of any elliptic curve, then p = 2,3,5,7,11, and that there is a bound B on the size of the group. I think (but am not sure) that B has been explicitly calculated. This was later generalized by Merel in "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" Invent. Math. 124 (1996), no. 1-3, 437--449, to the following: if K is a number field of degree d, then denote by S(d) the set of primes dividing the orders of any torsion subgroup of an elliptic curve over K. Then S(d) is bounded and the orders are bounded. You might look in Kamienny, Sheldon "Some remarks on torsion in elliptic curves", Comm. Algebra 23 (1995), no. 6, 2167--2169. Victor Miller -- CCR, Princeton ============================================================================== From: victor@idaccr.org (Victor S. Miller) Subject: Re: Possible Torsion on Elliptic Curves Date: 19 Feb 1999 11:58:20 -0500 Newsgroups: sci.math,sci.math.research You might look at http://xxx.lanl.gov/abs/alg-geom/9604003, a paper by Pierre Parent: Bornes effectives pour la torsion des courbes elliptiques sur les corps des nombres Abstract: We give an effective form of the theorem of Mazur-Kamienny-Merel on the torsion of elliptic curves over number fields. -- Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly victor@idaccr.org | be expected to keep writing papers saying 'I can do the CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed 08540 USA | what editor would publish them?" -- Oliver Atkin ============================================================================== From: Roberto Maria Avanzi Subject: Further Questions (was Re: Possible Torsion on Elliptic Curves) Date: Fri, 19 Feb 1999 14:03:33 +0100 Newsgroups: sci.math,sci.math.research On Thu, 18 Feb 1999, John R Ramsden wrote: > Roberto Maria Avanzi wrote in message ... > > > > Which possible groups appear as groups of rational torsion > > points on Elliptic Curves over the field Q(\sqrt{-7}) ? > > My cat is sitting on the pile of books that includes "Elliptic Curves" > by Silverman (Springer-Verlag), and I daren't disturb her. But Barry > Mazur classified all possible torsion groups of rational points of an > elliptic curve over Q into a finite number of cases. I know this, actually I used this result in a joint paper which has been recently submitted. > 11 seems to ring > a bell. His proof also produced a parametrization of each case. If > you track down this result it might be possible to extend it to > Q(\sqrt{-7}), although I'd guess this is a very difficult problem. It is a VERY tough problem. AFAIK there is a generalization to generic quadratic fields saying only which primes occur in the order of rational torsion points. Kamienny I think did the work.. Quastion: Does anybody know where a proof is published ? (I have only seen a research announcement). I never counted the number of cases, let's do it now, if T is the torsion subgroup of E(Q), then T is either isomorphic to C_2 x C_n with n = 2,4,6,8 or to C_n with 1 <= n <= 10 or n = 12. Should be 15 nonisomorphic groups. 11 is the smallest prime not occurring. IIRC the modular curve parametrizing curves with a (non-necessarily rational) point of order 11 is also elliptic (in the Silverman the curve is also given), but for order strictly less than 11 or equal to 10 the genus is 0 and that allows for nice parametrizations. Question: is the genus of this curve for n>=13 always bigger than 0 ? (I think I read it somewhere it is so). But my original question was anyway over Q(sqrt(-7)), a field which pops up insistently in some diophantine questions (and also related to polynomials), and is therefore the second most important field in my work (after Q). Maybe I will ask Noam Elkies (who is also a composer, how nice, I compose too) and Silverman... Thank you Roberto _/_/ Roberto Maria Avanzi /_/ Institut für Experimentelle Mathematik / Universität Essen _/ Ellernstraße 29 / 45326 Essen / Germany / Phone: ++49-201-183-7641 Fax: ++49-201-183-7668 ============================================================================== From: "Noam D. Elkies" Subject: Re: Possible Torsion on Elliptic Curves Date: 19 Feb 1999 20:41:55 GMT Newsgroups: sci.math,sci.math.research In article , Victor S. Miller wrote: >The problem of boundedness of torsion for elliptic curves over >quadratic fields was settled by Sheldon Kamienny in a series of >papers. He showed that if K is ANY quadratic number field, if p is a >prime dividing the order of the torsion subgroup of any elliptic >curve, then p = 2,3,5,7, [or] 11, This cannot be correct as stated because the curve X_1(13) parametrizing elliptic curves with a 13-torsion point has genus 2 -- in fact it is known that y^2 = x^6 + 2 x^5 + x^4 + 2 x^3 + 6 x^2 + 4 x + 1 is an equation for that curve. The six rational points at x=0,-1,inf are cusps, and any other rational x yields a non-cusp point defined over a quadratic extension K of Q, and thus an elliptic curve over K with a K-rational 13-torsion point. I do not know whether K can be Q(sqrt(-7)); if it can then it will only occur finitely often [Faltings]. But p=11 does occur infinitely often, because according to the Cremona tables the twist of the elliptic curve X_1(11) by Q(sqrt(-7)) has positive rank. I do not know whether the full list of possible torsion groups for Q(sqrt(-7)) has been determined. --Noam D. Elkies (elkies@math.my_university.edu) Dept. of Mathematics, Harvard University