A major open problem in arithmetic geometry is the Birch and Swinnerton-Dyer conjecture. An elliptic curves is an algebraic curve with at least one point with coordinates in such that the set of complex points is a topological torus. The Birch and Swinnerton-Dyer conjecture gives a criterion for whether or not is infinite in terms of analytic properties of the -function .
William Stein 2012-09-24