This article does not represent precisely a talk given at the symposium, but is complementary to [DenS]. Its purpose is to explain a setting in which the various conjectures on special values of L-functions admit a unified formulation. At critical points, Deligne’s conjecture [Del2] relates the value of an L-function to a certain period, and at non-critical points, the conjectures of Beilinson ...

Introduction. Although it has occupied a central place in number theory for almost a century, the arithmetic of elliptic curves is still today a subject which is rich in conjectures, but sparse in definitive theorems. In this lecture, I will only discuss one main topic in the arithmetic of elliptic curves, namely the conjecture of Birch and Swinnerton-Dyer. We briefly recall how this conjecture...

The theory of canonical heights on abelian varieties originated with the work of Néron [10] and Tate (first described in print by Manin [8]) in 1965. Tate’s simple and elegant limit construction uses a Cauchy sequence telescoping sum argument. Néron’s construction, which is via more delicate local tools, has proven to be fundamental for understanding the deeper properties of the canonical heigh...

In a few earlier papers ([8], [9], [10]) attention was called to the striking parallel between the ideas surrounding the well-known conjecture of Birch and Swinnerton-Dyer for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for ‘effective non-abelian descent’ on hyperbolic curves equipped ...

Let E/Q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of p-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when E(K) is infinite, and he proved that his conjecture is tru...

We review some of Kolyvagin’s results and conjectures about elliptic curves, then make a new conjecture that slightly refines Kolyvagin’s conjectures. We introduce a definition of finite index subgroups Wp ⊂ E(K), one for each prime p that is inert in a fixed imaginary quadratic field K. These subgroups generalize the group ZyK generated by the Heegner point yK ∈ E(K) in the case ran = 1. For a...

In this paper we present a method for explicitly computing rational points on elliptic curves using Heegner points. This method was crucial to the proof of Gross-Zagier, which proves the rank one case of the Birch and Swinnerton-Dyer Conjecture. Although this use of Heegner points can be found in many books and articles, we strive here to present it in a more concrete and complete form, and usi...