### Rank growth of elliptic curves over N-th root extensions

Linfoot Number Theory Seminar

1st June 2022, 11:00 am – 12:00 pm

Fry Building, Room 2.04

Fix an elliptic curve E/\mathbb{Q}. If \mathcal{K} is some family of finite extensions of \mathbb{Q}, then, for each field K\in\mathcal{K}, E(K) is a finitely generated abelian group. It is natural to ask how the rank of E(K) varies as K runs through \mathcal{K}. For example, if \mathcal{K} is the set of all quadratic extensions of \mathbb{Q}, then a conjecture of Goldfeld states that \mathrm{rk}\ E(K) = \mathrm{rk}\ E(\mathbb{Q})for 50\% of K, and \mathrm{rk}\ E(K) = \mathrm{rk}\ E(\mathbb{Q})+1 for the other 50\%. On the other hand, if \mathcal{K} is the set of all extensions of the form \mathbb{Q}(\sqrt[3]d), then, by work of V. Dokchitser, assuming BSD, there exist elliptic curves E such that \mathrm{rk}\ E(K) > \mathrm{rk}\ E(\mathbb{Q}) for all K\in\mathcal{K}.

In this talk, we will study the growth of the rank of E after base change to the fields K_d=\mathbb{Q}(\sqrt[N]d), where N=2\cdot3^m for some fixed m\ge 1. Our main result is that, if E admits a 3-isogeny, then the average "new rank" of E(K_d) is bounded as |d|\to\infty.

I will show how to deduce this theorem from a more general result about ranks of abelian varieties in cyclotomic twist families. If time permits, I'll discuss the proof of this more general result, which exploits a connection between 3-isogeny Selmer groups and binary cubic forms. This is joint work with Ari Shnidman.

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