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

Daniel Keliher

Vol. 331 (2024), No. 2, 331–352

DOI: 10.2140/pjm.2024.331.331

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 $\mathrm{rk}<!--FUNCTION\; APPLICATION-->(E∕K)-\mathrm{rk}<!--FUNCTION\; APPLICATION-->(E∕\mathbb{Q})$ for fixed $E$ and varying $K$. When $\mathrm{rk}<!--FUNCTION\; APPLICATION-->(E∕\mathbb{Q})\le 1$, 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 $\mathrm{rk}<!--FUNCTION\; APPLICATION-->(E∕K)-\mathrm{rk}<!--FUNCTION\; APPLICATION-->(E∕\mathbb{Q})=0$. 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 $\mathbb{Q}$.

Keywords

rank growth, elliptic curve, Selmer group, arithmetic statistics

Mathematical Subject Classification

Primary: 11G05

Milestones

Received: 24 May 2023

Revised: 19 September 2024

Accepted: 27 September 2024

Published: 30 October 2024