Log : bsd.tex, v Revision 1.26 2000/05/21 03:05:38 was bsd Revision 1.25 2000/05/15 07:31:56 was added I. Revision 1.24 2000/05/15 07:28:22 was Changed F_{p}toF_{v}inproofofmaintheorem! Revision 1.23 2000/05/15 05:11:14 was quick fix Revision 1.22 2000/05/15 04:46:48 was away from level. Revision 1.21 2000/05/15 04:46:08 was got rid of a_{p} Revision 1.20 2000/05/15 04:45:32 was third fix; added Weierstrass equation but didn’t change subse-

quent text!

Revision 1.19 2000/05/15 04:44:23 was quick fix: typo 2; geometric component group of F! Revision 1.18 2000/05/15 04:43:26 was quick fix. a_{p}typo Revision 1.17 2000/05/12 01:52:07 was oops! Revision 1.16 2000/05/10 20:32:49 was done. Revision 1.15 2000/05/10 09:39:43 was weakend explantory claim. Revision 1.14 2000/05/10 09:35:36 was addedMazur Revision 1.13 2000/05/10 09:33:08 was eliminated toric reduction Revision 1.12 2000/05/10 07:12:25 was Added index’s to tables. Revision 1.11 2000/05/09 20:21:57 was added a label to subsecNotation Revision 1.10 2000/05/08 15:42:17 was fixed up index some. Revision 1.9 2000/05/08 02:52:01 was Fixed mispelling of ”isogeneous”. Revision 1.8 2000/05/08 02:00:23 was Fixed up 1913. Made lots and lots of refinements and clarifcations

of the tables.

Revision 1.7 2000/05/07 21:57:57 was Added cremona’s model for 5389A. Revision 1.6 2000/05/07 19:51:10 was Added a section on when Sha becomes visible. Revision 1.5 2000/05/06 20:55:41 was Worked on cleaning up the descriptions of the tables. The Birch and Swinnerton-Dyer conjecture chap:bsdBirch and Swinnerton-Dyer conjecture—seeBSD

conjectureConjecture!Birch and Swinnerton-Dyer—seeBSD conjectureNow that the Shimura-TaniyamaShimura-Taniyama conjectureConjecture!Shimura and Taniyama conjecture has been proved, many experts consider the Birch and Swinnerton-Dyer conjecture (BSD conjecture) to be one of the main outstanding problems in the field (see [pg. 549]darmon-bsd and [Intro.]cime-1997). This conjecture ties together many of the arithmetic and analytic invariants of an elliptic curve. At present, there is no general class of elliptic curves for which the full BSD conjectureBSD conjecture!is still unknown is known, though a slightly weakened form is known for a fairly broad class of complex multiplication elliptic curves of analytic rank 0 (see rubin:main-conjectures), and several deep partial results have been obtained during the last twenty years (see, e.g., gross-zagier and kolyvagin:mordellweil).

Approaches to the BSD conjectureBSD conjecture that rely on congruences betweenCongruences!and BSD conjecture modular formsModular forms!and BSD are likely to require a deeper understanding of the analogue of the BSD conjectureBSD conjecture!in higher dimensions for higher-dimensional abelian varieties. As a first step, this chapter presents theorems and explicit computations of some of the arithmetic invariants of modular abelian varieties.

The reader is urged to also read A. Agashe’s 2000 Berkeley Ph.D. thesis which cover similar themes.Agashe The paper of Cremona and Mazur’s cremona-mazurMazur paints a detailed experimental picture of the way in which congruences link Mordell-Weil and Shafarevich-Tate groups of elliptic curves. Congruences!and BSD conjecture

The BSD conjectureBSD conjecture—textit By breuil-conrad-diamond-taylor we now know that every elliptic curve over is a quotient of the curve X_{0}(N), whose complex points are the isomorphism classes of pairs consisting of a (generalized) elliptic curve and a cyclic subgroup of order N. Let J_{0}(N) denote the JacobianJacobian of X_{0}(N); this is an abelian variety of dimension equal to the genus of X_{0}(N) whose points correspond to the degree 0 divisor classes on X_{0}(N). The survey article diamond-im is a good guide to the facts and literature about the family of abelian varieties J_{0}(N).

Following Mazur mazur:rationalMazur, we make the following definition. definition[Optimal quotient]Optimal quotient—textit An optimal quotient of J_{0}(N) is a quotient A of J_{0}(N) by an abelian subvariety.