Numerical Evidence for the Equivariant Birch and Swinnerton-Dyer Conjecture

Modular symbols and the ( equivariant ) conjecture of Birch and Swinnerton - Dyer chris wuthrich 8 th July 2011

An Introduction to the Birch and Swinnerton-Dyer Conjecture

This article explores the Birch and Swinnerton-Dyer Conjecture, one of the famous Millennium Prize Problems. In addition to providing the basic theoretic understanding necessary to understand the simplest form of the conjecture, some of the original numerical evidence used to formulate the conjecture is recreated. Recent results and current problems related to the conjecture are given at the en...

The Birch-swinnerton-dyer Conjecture

We give a brief description of the Birch-Swinnerton-Dyer conjecture which is one of the seven Clay problems.

Visible Evidence for the Birch and Swinnerton-dyer Conjecture for Modular Abelian Varieties of Analytic Rank Zero Amod Agashe and William Stein, with an Appendix by J. Cremona and B. Mazur

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af , 1)/ΩAf , develop tools for computing with Af , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af ha...

On the elliptic curves of the form $ y^2=x^3-3px $

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.