Cambridge Journal of Mathematics
Volume 2 (2014)
Congruent numbers and Heegner points
Pages: 117 – 161
A positive integer is called a congruent number if it is the area of a right-angled triangle, all of whose sides have rational length. The problem of determining which positive integers are congruent is buried in antiquity (see Chapter 9 of Dickson), with it long being known that the numbers 5, 6, and 7 are congruent. Fermat proved that 1 is not a congruent number, and similar arguments show that also 2 and 3 are not congruent numbers. No algorithm has ever been proven for infallibly deciding whether a given integer $n \geq 1$ is congruent. The reason for this is that it can easily be seen that an integer $n \geq 1$ is congruent if and only if there exists a point $(x, y)$, with $x$ and $y$ rational numbers and $y \ne 0$, on the elliptic curve $ny^2 = x^3 - x$. Moreover, assuming $n$ to be square free, a classical calculation of root numbers shows that the complex $L$-function of this curve has zero of odd order at the center of its critical strip precisely when n lies in one of the residue classes of 5, 6, or 7 modulo 8. Thus, in particular, the unproven conjecture of Birch and Swinnerton-Dyer predicts that every positive integer lying in the residue classes of 5, 6, and 7 modulo 8 should be a congruent number. The aim of this paper is to prove the following partial results in this direction.