The Birch--Swinnerton-Dyer conjecture: beyond dimension 1

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One of the central problems of number theory is to describe the rational solutions of polynomial equations. The case of linear and quadratic equations is elementary, but the case of cubic equations (so-called "elliptic curves") is vastly deeper. The study of elliptic curves is a major strand of number theory, with connections to many other disciplines including topology, geometry, and cryptology. For an elliptic curve, the set of rational points on the curve forms an abelian group, which is known to be finitely generated. We define the "rank" of the curve to be the rank of this group; the central problem in the theory of elliptic curves is how to determine this rank. The Birch and Swinnerton-Dyer conjecture, one of the most famous open problems in mathematics, gives a conjectural formula for this rank, relating it to an auxiliary object known as an L-function.The goal of this project is to make new progress on the Birch--Swinnerton-Dyer conjecture for elliptic curves, and its generalisations to higher-dimensional objects ('abelian varieties'), using mathematical tools known as Euler systems.