Analytic ranks of elliptic curves over number fields

TL;DR

This paper investigates the average analytic ranks of elliptic curves over various number fields, establishing bounds that are independent of the extension degree and providing new insights into their distribution.

Contribution

It introduces bounds on the average analytic rank of elliptic curves over cyclic and $S_d$-field extensions, extending understanding beyond base fields.

Findings

Average analytic rank over cyclic extensions is at most 2 plus the base rank.

Bound on average rank is independent of the extension degree.

Provides new results on average ranks over $S_d$-fields.

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, we show that the average analytic rank of $E$ over cyclic extensions of degree $l$ over $\mathbb{Q}$ with $l$ a prime not equal to $2$, is at most $2+r_{\mathbb{Q}}(E)$, where $r_{\mathbb{Q}}(E)$ is the analytic rank of the elliptic curve $E$ over $\mathbb{Q}$. This bound is independent of the degree $l$ Also, we also obtain some average analytic rank results over $S_d$-fields.