Explicit Approaches to the Birch and Swinnerton-Dyer Conjecture

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ABSTRACT for award DMS-0653968 of SteinThe Birch and Swinnerton-Dyer conjecture is one of the most central unsolved problems in number theory. The proposed project aims to study this conjecture from two perspectives. First, the PI intends to continue his program to explicitly verify the full conjecture in many specific cases using explicit computations with Iwasawa theory, Euler systems, L-functions, and Galois cohomology; this work simultaneously motivates the development of new algorithms and the refinement of existing theorems. Second, the PI plans to carry out a theoretical and computational study of properties of Kolyvagin's Euler system of Heegner points, with the hope of carrying forward several of the ideas Kolyvagin left unfinished in the early 1990s.Elliptic curves play a central role in modern number theory and arithmetic geometry. For example, Andrew Wiles proved Fermat's Last Theorem by showing that the elliptic curve attached by Gerhard Frey to a counterexample to Fermat's claim would be attached to a modular form, and that this modular form cannot exist. Our understanding of the world of elliptic curves is extensive, but many questions remain unresolved. Perhaps the most central unsolved problem is the Birch and Swinnerton-Dyer conjecture, which relates many of the arithmetic invariants of an elliptic curve. The goals of this project are to provide substantial new data, techniques, and ideas relevant to attacks on the Birch and Swinnerton-Dyer conjecture.