Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals

TL;DR

This paper studies how the rank of elliptic curves over the rationals changes when extended to certain quartic fields with Galois groups $S_4$ and $A_4$, showing conditions under which the rank remains unchanged.

Contribution

It proves the existence of infinitely many $S_4$ quartic extensions where elliptic curves with rank at most 1 do not gain rank, by controlling the 2-Selmer rank in quadratic extensions.

Findings

Existence of infinitely many $S_4$ extensions with unchanged rank for certain elliptic curves.

Method to control 2-Selmer rank in quadratic extensions.

Insights into rank behavior in quartic Galois extensions.

Abstract

We investigate the rank growth of elliptic curves from $\mathbb{Q}$ to $S_4$ and $A_4$ quartic extensions $K/\mathbb{Q}$. In particular, we are interested in the quantity $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q})$ for fixed $E$ and varying $K$. When $\mathrm{rk}(E/\mathbb{Q}) \leq 1$, with $E$ subject to some other conditions, we prove there are infinitely many $S_4$ quartic extensions $K/\mathbb{Q}$ over which $E$ does not gain rank, i.e. such that $\mathrm{rk}(E/K) - \mathrm{rk}(E/\mathbb{Q}) = 0$. To do so, we show how to control the 2-Selmer rank of $E$ in certain quadratic extensions, which in turn contributes to controlling the rank in families of $S_4$ and $A_4$ quartic extensions of $\mathbb{Q}$.