We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the L-series is true for...

Let E be an optimal elliptic curve of conductor N , such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from...

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.

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

The paper describes an approach that may lead to a proof that the Birch and Swinnerton-Dyer conjecture fails.

Abstract This article proves a case of the p -adic Birch and Swinnerton-Dyer conjecture for Garrett L -functions [6], in exceptional zero setting extended analytic rank 2.

Abstract We show that the conjecture of [27] for special value at $s=1$ zeta function an arithmetic surface is equivalent to Birch–Swinnerton–Dyer Jacobian generic fibre.