On quadratic families of CM elliptic curves

Munshi, Ritabrata

doi:10.1090/S0002-9947-2011-05433-4

On quadratic families of CM elliptic curves
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Trans. Amer. Math. Soc. 363 (2011), 4337-4358 Request permission

Abstract:

Given a CM 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 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 $|u|, |v| \leq X$ is at least $X^{2-\varepsilon }$ for any $\varepsilon >0$.

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