Upper bounds for analytic ranks of elliptic curves over cyclotomic fields

TL;DR

This paper establishes new upper bounds for the analytic ranks of elliptic curves over cyclotomic fields, improving previous bounds and advancing understanding of their growth behavior over these extensions.

Contribution

The paper provides a significantly improved upper bound for the analytic rank of elliptic curves over cyclotomic fields, refining prior results by Chinta.

Findings

Bound on analytic rank is improved to q^{45/52+ε}

Demonstrates growth rate of analytic ranks over cyclotomic extensions

Advances theoretical understanding of elliptic curve ranks in number theory

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 $q^{45/52+\varepsilon}$, as $q\to \infty$ through the primes. This improves the bound $q^{7/8+\varepsilon}$ established by Chinta.