To a large extent, the investigations to be brought up today arise from a curious inadequacy having to do with the arrow on the left. On the one hand, it is widely acknowledged that the theory of motives finds a strong source of inspiration in Diophantine geometry, inasmuch so many of the structures, conjectures, and results therein have as model the conjecture of Birch and Swinnerton-Dyer, whe...

We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies ∑ f∈S2(q)∗ ords= 1 2 L(f, s) ≤ C|S2(q) ∗| with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies rank J0(q) ≤ C dim J0(q) for q prime large enough.

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptoti...

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...

For primes $$q \equiv 7 \ \mathrm {mod}\ 16$$ , the present manuscript shows that elementary methods enable one to prove surprisingly strong results about Iwasawa theory of Gross family elliptic curves with complex multiplication by ring integers field $$K = {\mathbb {Q}}(\sqrt{-q})$$ which are in perfect accord predictions conjecture Birch and Swinnerton-Dyer. We also some interesting phenomen...

Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...

This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and L-functions” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of MordellW...