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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.09497 |
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Table of Contents:
- 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 $\mathbb{Q}$.