2/8/2012
S. Zhang
Princeton University
Congruent numbers and Heegner points
An anonymous Arab manuscript, written before 972, contains a `` problem of congruent numbers": given an integer n, to find a rational square x^2 such that x^2+n and x^2-n are both rational squares. For example 1, 2, 3 are not congruent numbers but 5, 6, 7 are. A modern equivalence of this problem is to find a point with infinite order on the elliptic curve: y^2=x^4-n^2. A special case of the Birch and Swinnerton--Dyer conjecture assets that any positive integer n=5, 6, 7 mod 8 are congruent number while almost all n=1, 2, 3 mod 8 are not congruent.