Ranks of Elliptic Curves Twisted by Quadratic Forms

Mohammad H. Hamdar; Cihan Sabuncu

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arXiv · 2607.13000

Ranks of Elliptic Curves Twisted by Quadratic Forms

Mohammad H. Hamdar·Cihan Sabuncu

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Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $E^d$ be its twist by the quadratic character $\chi_d$. 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.

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Mohammad H. Hamdar, Cihan Sabuncu. 2026-07-14. Ranks of Elliptic Curves Twisted by Quadratic Forms. https://arxiv.org/abs/2607.13000

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