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

TL;DR

This paper investigates how the Mordell--Weil rank of elliptic curves over certain root extensions grows, showing boundedness in some cases and no new points in others, with implications for Hilbert's tenth problem.

Contribution

It introduces the correlation trick technique and applies it to study rank growth and rational points in root extensions, extending previous results.

Findings

Average new rank is bounded for elliptic curves with a 3-isogeny over $K_d$.

Many elliptic curves over $Q$ have no new points over $Q( oot 6 d)$ for a positive proportion of $d$.

Hilbert's tenth problem is negatively solvable over many pure sextic fields.

Abstract

Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$. We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $K_d = F(\sqrt[2n]{d})$. If $E$ admits a $3$-isogeny, then we show that the average ``new rank'' of $E$ over $K_d$, appropriately defined, is bounded as the height of $d$ goes to infinity. When $n = 3$, we moreover show that for many elliptic curves $E/\mathbb{Q}$, there are no new points on $E$ over $\mathbb{Q}(\sqrt[6]d)$, for a positive proportion of integers $d$. This is a horizontal analogue of a well-known result of Cornut and Vatsal. As a corollary, we show that Hilbert's tenth problem has a negative solution over a positive proportion of pure sextic fields $\mathbb{Q}(\sqrt[6]{d})$. The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the ``correlation trick'', which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.