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...

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...

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 ...

Abstract In a paper published in 1980, the author gave an adelic Tamagawa number interpretation for Birch and Swinnerton-Dyer conjecture divisors on abelian varieties. Some years later, joint work with K. Kato, more general volume zeta values of motives weights $${<}\,{-1}$$ was proposed. at hand, is generalized to deal weight $$-1$$ . As points varieties are replaced by cohomology coefficie...

Abstract The purpose of this paper is to give a formula for the leading coefficient at s = 1 {s=1} L -function one-motives over function fields in terms Weil-étale cohomology, generalizing version Birch and Swinnerton-Dyer conjecture authors’ previous work. As consequence we...

We prove the Gross–Zagier–Zhang formula over global function fields of arbitrary characteristics. It is an explicit which relates Néron-Tate heights CM points on abelian varieties and central derivatives associated quadratic base change L-functions. Our proof based arithmetic variant a relative trace identity Jacquet. This approach proposed by Zhang. apply our results to Birch Swinnerton–Dyer c...

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of N−1 12 . Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the s...