Constructing curves of high rank via composite polynomials

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.