Ranks of Elliptic Curves Twisted by Quadratic Forms

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $E^d$ be its twist by the quadratic character $χ_d$. We prove there are infinitely many twists $d$ which are sums of two squares such that $E^d$ has rank $1$. This result is achieved using moments of derivatives of modular $L$-functions, and particularly captures the lower derivatives which were left out in the work of Munshi. Such a result, in particular, also gives us information on the elliptic fibration $(1+t^2)y^2=f(x)$, where $f(x)$ is a cubic polynomial.