Rank-one twists of a certain elliptic curve

Abstract

The purpose of this note is to give the first known example for a conjecture of Goldfeld on the number of rank-one curves appearing in a family of quadratic twists. We show unconditionally that the curve X0(19) has the property that a positive proportion of its quadratic twists have analytic rank one. This amounts to a strong nonvanishing statement for the derivatives of certain L-functions when the sign of the functional equation is −1. We simultaneously obtain the fact that a positive proportion of twists have analytic rank zero when the sign of the functional equation is +1. Let E be a modular elliptic curve over Q, with corresponding newform F (z) = ∑ anq n ∈ S2(Γ0(N)). Let L(s, F ) = L(s, E) = ∑ ann −s be the usual L-series. Then L(s, F ) satisfies a functional equation under s → 2− s, with sign = ±1 = (E). Let D be the fundamental discriminant of a quadratic field K = K( √ D) and write χD for the associated quadratic character. In this setting we define the twisted L-function L(s, FD) = ∑ χD(n) · ann. If E is given by the equation Y 2 = P (X), then L(s, FD) is the L-function attached to the twisted elliptic curve ED given by DY 2 = P (X). Therefore L(s, FD) also satisfies a functional equation and, if (D,N) = 1, then the sign is given by D = · χD(−N). For r = 0, 1, and a positive real number X, define M r F (X) = #{D : |D| < X : Ords=1L(s, FD) = r}. The following conjecture of Goldfeld [Gol79] is well-known: