---
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.'
address:
- 'Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA'
- 'Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA'
- 'Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA'
author:
- 'Harris B. Daniels'
- 'Álvaro Lozano-Robledo'
- Erik Wallace
bibliography:
- 'Alvaro-Harris-Erik.bib'
title: 'Bounds of the rank of the Mordell–Weil group of Jacobians of Hyperelliptic Curves'
---

Introduction
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Let $C/{\mathbb Q}$ be a hyperelliptic curve, given by a model $y^2=f(x)$, with $f(x)\in{\mathbb Q}[x]$, and let $J/{\mathbb Q}$ be the jacobian variety attached to $C$. The Mordell–Weil theorem shows that $J({\mathbb Q})$ is a finitely generated abelian group 