Upper bounds for analytic ranks of elliptic curves over cyclotomic fields

TL;DR

This paper studies the growth of the analytic rank of an elliptic curve $E/\mathbb{Q}$ over cyclotomic fields $K_q=\mathbb{Q}(e^{2\pi i/q})$ as $q\to\infty$. It develops a mollified average over Galois orbits of Dirichlet twists $L(E\otimes χ, 1/2)$ to obtain an unconditional bound, improving the previous exponent from $q^{7/8+\varepsilon}$ to $q^{45/52+\varepsilon}$. A general non-vanishing result is established for twists of modular forms, showing $L(f\otimes χ, 1/2) \neq 0$ when the order of $χ$ is larger than $q^{45/52+\varepsilon}$. The key technical advance is a refined cancellation analysis in sums of Kloosterman sums, enabling control of the dual sum in the approximate functional equation and advancing unconditional non-vanishing results in high-order twist families.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We show that the analytic rank of $E$ over the cyclotomic extension $\mathbb{Q}(e^{2πi/q})$ 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.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Proposition 3.1
• Lemma 4.1
• proof
• Lemma 4.2