Curves Dy 2 = x 3 — x of Odd Analytic Rank

Abstract

For nonzero rational D, which may be taken to be a square-free integer, let E d be the elliptic curve Dy 2 = x 3 — x over Q arising in the “congruent number” problem.1 It is known that the L-function of E d has sign —1, and thus odd analytic rank r an(Ed), if and only if |D| is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and Swinnerton-Dyer that the arithmetic rank of each of these curves E d is odd, and therefore positive. We prove that E d has positive rank for each D such that |D| is in one of the above congruence classes mod 8 and also satisfies |D| < 106. Our proof is computational: we use the modular parametrization of E 1 or E 2 to construct a rational point P d on each E d from CM points on modular curves, and compute P d to enough accuracy to usually distinguish it from any of the rational torsion points on E d . In the 1375 cases in which we cannot numerically distinguish P d from (E d )tors, we surmise that P d is in fact a torsion point but that E d has rank 3, and prove that the rank is positive by searching for and finding a non-torsion rational point. We also report on the conjectural extension to |D| < 107 of the list of curves E d with odd r an (E d ) > 1, which raises several new questions.