Upper bounds for analytic ranks of elliptic curves over cyclotomic fields

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We show that the analytic rank of $E$ over the cyclotomic extension $\mathbb{Q}(e^{2\pi i/q})$ is bounded above by $q^{45/52+\epsilon }$, as $q\to \infty$ through the primes. This improves the bound $q^{7/8+\epsilon }$ established by Chinta.