2002/08/07 by Noam D. Elkies
Mathematics · #math.NT #msc:14H52 #msc:11G15 #msc:11Y50 #msc:11Y70
published as Pages 244-251 of the proceedings of ANTS-V (LNCS 2369 (2002), ed. C.Fieker and D.Kohel) · 8 pages, 1 figure
arxiv created 2002/08/07 · arxiv updated 2009/11/30
For nonzero rational D, which may be taken to be a squarefree integer, let ED be the elliptic curve Dy2=x3-x over Q arising in the "congruent number" problem. It is known that the L-function of ED has sign -1, and thus odd analytic rank, if and only if |D| is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and Swinnerton-Dyer that the arithmetic rank of each of these curves ED is odd, and therefore positive. We prove that ED has positive rank for each D such that |D| is in one of the above congruence classes mod 8 and also satisfies |D|<106. Our proof is computational: we use the modular parametrization of E1 or E2 to construct a rational point PD on each ED from CM points on modular curves, and compute PD to enough accuracy to usually distinguish it from any of the rational torsion points on ED. In the 1375 cases in which we cannot numerically distinguish PD from a torsion point of ED, we surmise that PD is in fact a torsion point but that ED has rank 3, and prove that the rank is positive by searching for and finding a non-torsion rational point. We also report on the conjectural extension to |D|<107 of the list of curves ED whose analytic rank is odd and greater than 1, which raises several new questions.