Constructing curves of high rank via composite polynomials

Arvind Suresh

Published 2021-02-03Version 1

We improve on a construction of Mestre--Shioda to produce some families of curves $X/\mathbb{Q}$ of record rank relative to the genus $g$ of $X$. Our first main result is that for any integer $g \geqslant 8$ with $g \equiv 2 \pmod 3$, there exist infinitely many genus $g$ hyperelliptic curves over $\mathbb{Q}$ with at least $8g+32$ $\mathbb{Q}$-points and Mordell--Weil rank $\geqslant 4g + 15$ over $\mathbb{Q}$. Our second main theorem is that if $g+1$ is an odd prime and $K$ contains the $g+1$-th roots of unity, then there exist infinitely many genus $g$ hyperelliptic curves over $K$ with Mordell--Weil rank at least $6g$ over $K$.