[12pt]article macros [all]xy Nonvanishing Twists and Visible Shafarevich-Tate Groups October 29, 2001 at Boston University William Stein http://modular.fas.harvard.edu Let be an elliptic curve over and let be a prime. One of my long-term goals is to find a rank modular abelian varieties that fits into a short exact sequence in such a way that the exact sequence of Galois cohomology induces an isomorphism I am searching for such because I hope to relate the rank Birch and Swinnerton-Dyer formula to the conjecture that if and only if is infinite. 0.1ex Acknowledgement: Stoll, Chinta, Rohrlich, Gross, Mazur, and Poonen. Terminology Let denote the conductor of , and for each prime let denote the component group of the Neron model of at . A prime is rigid for if and the Galois representation is irreducibleMazur proved in 1978 that if then is irreducible.. A Conjecture About Nonvanishing of Twists Suppose is rigid for . For every , let be a Dirichlet character of order and conductor . [--] For every rigid prime , there exists a prime with such that and Evidence: This conjecture is true for every pair with rigid and any of the first rank-one optimal quotients of or either of the first rank-two optimal quotients. Also, people such as David Rohrlich and Gautam Chinta have proved similar statements, and Hoffstein is rumored to be working on a very similar question. Jack Fearnley of Montreal proved something similar in his thesis in the case when . Avoiding -torsion Let be a rigid prime for , and suppose there is a prime with the properties predicted by Conjecture . Let be the abelian extension corresponding to . The restriction of scalars is an abelian variety over of dimension ; it represents the function . There is a natural injection and a natural surjection with connected kernel. If we let then since the composite is multiplication by , the setup is as follows: If , then all have trivial -torsion. The action of the Frobenius automorphism on has characteristic polynomial Our hypothesis on implies that this polynomial does not have as a root, so We next prove that By definition, where is the completion of at the unique prime over . Also, because , we have . If then , so since is totally ramified, , so . It follows that . To see that note that since is and the diagonal embedding splits. Finally, consider . Let denote the Neron model of over . By Lang's Lemma, Thus if , then . Since , Hensel's lemma (and formal groups) imply that , contrary to the fact that . Visualizing Mordell-Weil in Rank 0 Sha Let be a closed immersion of abelian varieties. Then Barry Mazur defined Suppose and is finite as a set. If is a prime such that and then (Note: If is infinite, then no primes satisfy the hypothesis of the theorem.) Amod Agashe and I prove most of this theorem in our paper Visibility of Shafarevich-Tate Groups of Abelian Varieties. The proof uses the snake lemma and a careful local analysis at each prime. Let be an elliptic curve over . Conjecture implies that for every rigid prime , there is an abelian extension of degree such that where and has dimension and rank . Conjecture produces a prime with such that and . We will apply Theorem to , , and where is the abelian extension corresponding to and . Since and is attached to the modular form , Kato's work implies that is finite, so is finite. Lemma implies that The only hypothesis of Theorem left to check is that for all . This is because since is unramified at (since ), formation of Neron models commutes with tame base extensions, and is rigid for . Thus Theorem implies that A Connection with the BSD Conjecture Let be an elliptic curve and suppose that . Suppose that (1) we can prove that there is a rigid prime for and a prime as in Conjecture . Then (2) the rank BSD formula would predict that sinceI actually haven't written down a proof of this congruence in general, but it should be easy. and . If (3) we also have , then , so is infinite. Let be the rank elliptic curve 389A. The prime is rigid, and satisfies . We have for a rank abelian variety of dimension . It would be interesting to investigate the relationship between Kato's Euler system for and this subgroup of . A Nonsquare Shafarevich-Tate Group Let be the rank elliptic curve of conductor . Then is rigid and satisfies Conjecture . We have for the two-dimensional kernel of . Amazingly, the Birch and Swinnerton-Dyer conjecture predicts that , which is not a perfect square. This is the first abelian variety ever found in which is not a perfect square for an odd prime .