A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM

We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle theta. These numbers are called theta-congruent. We give an elliptic curve criterion for determining whether a given integer n is theta-congruent. We then consider the "density" of integers n which are theta-congruent, as well as the related problem giving the "density" of angles theta for which a fixed n is congruent. Assuming the Shafarevich-Tate conjecture, we prove that both proportions are at least 50% in the limit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.