Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function LE(s) of E does not vanish at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to E modulo a power r of a prime p. The theory of vis...

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...

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...

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...

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...

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...