Congruent Numbers and Heegner Points

S. Zhang, Princeton University
Fine Hall 314

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.