Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents

TL;DR

This paper investigates how often elliptic curves over a number field increase their rank when extended to certain S3 cubic fields with quadratic resolvents, providing probabilistic bounds on rank growth.

Contribution

It introduces a new distribution of Selmer ranks for elliptic curves over S3 extensions, establishing probabilistic bounds on rank gain.

Findings

Elliptic curves gain rank at most once over S3 extensions with probability at least 31.95%.

Distribution of Selmer ranks is determined under a non-standard ordering.

Provides probabilistic insights into rank growth in specific field extensions.

Abstract

We study the probability with which an elliptic curve $E/k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-cubic extension $K/k$ with quadratic resolvent field $F/k$, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to $E$ and $S_3$-cubic extensions $K/k$ following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that $E$ gains rank by at most one upon base extension to $K$ with probability at least $31.95\%$.