Congruent numbers over real quadratic fields

Abstract

Let m O01U be a square-free positive integer. We say that a positive integer n is a congruent number over QO AAAA p U if it is the area of a right triangle with three sides in QO AAAA p U. We put Ka QO AAAA p U. We prove that if m0 2, then n is a congruent number over K if and only if EnOKU has a positive rank, where EnOKU denotes the group of K-rational points on the elliptic curve En defined by y 2 a x 3 ˇ n 2 x. Moreover, we classify right triangles with area n and three sides in K.