Curves D y^2 = x^3 - x of odd analytic rank

Abstract

For nonzero rational D, which may be taken to be a squarefree integer, let E_D be the elliptic curve Dy^2=x^3-x over Q arising in the "congruent number" problem. It is known that the L-function of E_D has sign -1, and thus odd analytic rank, 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|<10^6. 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 a torsion point of E_D, 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|<10^7 of the list of curves E_D whose analytic rank is odd and greater than 1, which raises several new questions.