Efficient Verification of Tunnell's Criterion

Abstract

An integer $n$ is congruent if there is a triangle with rational sides whose area is $n$. In the 1980s Tunnell gave an algorithm to test congruence which relied on counting integral points on the ellipsoids $2x^2+y^2+8z^2 = n$ and $2x^2+y^2+32z^2=n$. The correctness of this algorithm is conditional on the conjecture of Birch and Swinnerton-Dyer. The known methods for testing Tunnell's criterion use $O(n)$ operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using time $n^{1/2 + o(1)}$ and space $n^{o(1)}$, and a deterministic algorithm using space and time $n^{1/2 + o(1)}$.