Binary quadratic forms and elliptic curves with analytic rank one

Abstract

Given an elliptic curve with Weierstrass equation $y^2=f(x)$, and a positive definite binary quadratic form $Q(u,v)$. We show that there are infinitely many $d$ in the set represented by the quadratic forms in the genus of $Q$ such that the twisted elliptic curve $d{y}^2=f(x)$ has analytic rank one.

13 pages