Constructing curves of high rank via composite polynomials

TL;DR

This paper refines a construction method to produce families of hyperelliptic curves over various number fields with record numbers of rational points and high Mordell--Weil rank relative to their genus, using classical polynomials.

Contribution

It introduces an improved construction of hyperelliptic curves with record rational points and ranks, extending previous methods and applying to larger number fields.

Findings

Constructed hyperelliptic curves with record rational points over various fields.

Achieved higher Mordell--Weil ranks relative to genus compared to previous records.

Connected classical Chebyshev polynomials to the defining equations of these curves.

Abstract

Let $k$ be a number field. We refine a construction of Mestre--Shioda to construct (infinite) families of hyperelliptic curves $X/{k}$ having a record number of rational points and record Mordell--Weil rank relative to the genus of $g$ of $X$. Over $k=\mathbb{Q}$, we obtain modest improvements on the current published records, and these improvements become more significant as $k$ gets larger. For example, we obtain curves over the real cyclotomic field $k = \mathbb{Q}(\cos(\pi/(g+1)))$ having at least $16(g+1)$ $k$-points and rank at least $8g$. The defining equations for the curves are closely related to the classical Chebyshev polynomials, and in special cases, we recover families studied (for example) by Mestre, Shioda, Brumer, and Tautz--Top--Verberkmoes.