On quadratic families of CM elliptic curves

Abstract

Given a CM elliptic curve with Weierstrass equation y 2 = f(x), and a positive definite binary quadratic form Q(v, v), we show that there are infinitely many reduced integer pairs (u, v) such that the twisted elliptic curve Q(u, v)y 2 = f(x) has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with |u|, |v| ≤ X is at least X 2―e for any e > 0.