Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves

TL;DR

This paper investigates bounds on the Mordell-Weil rank of jacobians of hyperelliptic curves over rationals, linking it to the genus and class group properties of associated cyclic fields, and provides examples where these bounds are tight.

Contribution

It extends previous work to establish bounds on the Mordell-Weil rank for hyperelliptic jacobians related to Sophie Germain primes, with explicit examples demonstrating sharpness.

Findings

Rank is bounded by the genus and class group 2-rank.

Examples show the bounds are sharp.

Connections to cyclic extensions and cubic fields.

Abstract

In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.