Euler's concordant forms

Abstract

This is known as Euler’s concordant forms problem, and when M = N Euler’s problem is the congruent number problem. Tunnell gave a conditional solution to the congruent number problem using elliptic curves and modular forms. Using these ideas, we consider Euler’s problem which reduces to a study of the elliptic curve over Q : EQ(M,N) : y 2 = x 3 + (M +N)x 2 +MNx. If EQ(M,N) has positive rank, then there are infinitely many primitive integer solutions to (1); but if EQ(M,N) has rank 0, then there may be a non-trivial solution. Such a solution exists if and only if the torsion group is Z2◊Z8 or Z2◊Z6. We classify all such cases, thereby reducing Euler’s problem to a question of ranks. In some cases, the ranks of quadratic twists of EQ(M,N) are described by the representations of integers by ternary quadratic forms. Consequently, we obtain results regarding Euler’s problem, and the existence of solutions to a pair of Pell’s equations. Moreover, we give a new and elementary method, using the theory of lacunary modular forms, which establishes that there are infinitely many rank 0 quadratic twists of EQ(M,N) by discriminants in arithmetic progressions.