We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields. We prove some conditional results for the p′-part on it, and prove the p′-part of the conjecture for constant or isoconstant Abelian schemes, in particular the p′-part for (1) relative elliptic curves with good reduction or ...

The purpose of these notes is to describe the notion of an Euler system, a collection of compatible cohomology classes arising from a tower of fields that can be used to bound the size of Selmer groups. There are applications to the study of the ideal class group, Iwasawa’s main conjecture, Mordell-Weil group of an elliptic curve, X (the Safarevich-Tate group), Birch-Swinnerton-Dyer conjecture,...

Abstract In the present paper, we generalize celebrated classical lemma of Birch and Heegner on quadratic twists elliptic curves over ℚ {{\mathbb{Q}}} . We prove existence explicit infinite families with analytic ranks 0 1 for a large class curves, use points to explicitly construct rational order rank 1. addition, sho...

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of special L-value 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 q divides the algebraic part of special L value of A, as we...

We make several conjectures concerning the relations between the orders of the torsion subgroup, the arithmetic component groups, and the cuspidal subgroup of an optimal elliptic curve. These conjectures have implications for the second part of the Birch and Swinnerton-Dyer conjecture.

Let p and q be two distinct primes and let Je denote the winding quotient at level pq. We give an explicit formula that expresses the special L-value of Je as a rational number, and interpret it in terms of the Birch-Swinnerton-Dyer conjecture.

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x + y = z in Q( √ d), conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.