The size of Selmer groups for the congruent number problem, II

The oldest problem in the theory of elliptic curves is to determine which positive integers D can be the common difference of a three term arithmetic progres-sion of squares of rational numbers. Such integers D are known as congruent numbers. Equivalently one may ask which elliptic curve

The rank of elliptic curves over real quadratic number fields of class number 1

Abstract. In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over Q. Several examples are included. 1