From dweger@wis.few.eur.nlFri Jun 7 09:35:26 1996 Date: Thu, 6 Jun 1996 12:49:06 -0400 From: Benne de Weger To: Multiple recipients of list NMBRTHRY Subject: Big Sha Recently I have obtained some results on elliptic curves (defined over Q) with large Tate-Shafarevich groups. Let N be the conductor, and S the order of Sha. Goldfeld and Szpiro (Compositio 97, 1995) conjectured that S << N^{1/2+epsilon} . My conjecture is that there exist elliptic curves with S >> N^{1/2-epsilon} . I can show that this conjecture follows from the following standard conjectures: the Birch - Swinnerton-Dyer Formula (I only need the rank 0 case, which is almost proved by Kolyvagin), the Szpiro Conjecture (Discr << N^{6+epsilon}), and a Riemann hypothesis for a certain modular form of weight 3/2. The proof uses Frey curves for good ABC-examples. Based on the idea of the proof I did some computational experiments. The most spectacular example I found is the curve y^2 + x y + y = x^3 + x^2 - 16272564754316406252451 x - - 798973042220714620227331980906826 , which has a (conjectured, ``analytical'') Sha of order S = 50176 (= 224^2), and conductor N = 51636585. So the ratio S / N^{1/2} is 6.983. A preprint will be available soon. Benne de Weger University of Leiden / Erasmus University Rotterdam new e-mail address: deweger@few.eur.nl old e-mail address: dweger@wis.few.eur.nl (still should work for a long time)