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

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 $\mathrm{rk}(E/\mathbb{Q})\le 1$, 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 $\mathrm{rk}(E/K)-\mathrm{rk}(E/\mathbb{Q})=0$. 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 $\mathbb{Q}$.

v2: 21 pages. Updates to exposition in sections 2 and 5. Slight improvement to the main theorem. Accepted by Pacific Journal of Mathematics

References in corpus (1)

• Rank growth of elliptic curves over $N$-th root extensions

Cited by in corpus (1)

• Selmer stability for elliptic curves in Galois $\ell$-extensions