Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two

TL;DR

The paper proves the existence of infinitely many hyperelliptic curves over any number field with Jacobians of fixed small rank, across various genera, demonstrating the abundance of such curves.

Contribution

It establishes the existence of infinitely many hyperelliptic curves with Jacobians of small fixed rank for any genus over any number field, extending previous results.

Findings

Infinitely many hyperelliptic curves of genus g with Jacobian rank 0, 1, or 2 over any number field.

Explicit examples over $ ext{Q}$ for ranks between 1 and 11 in genus 2.

Existence of infinitely many genus 3-6 hyperelliptic curves with Jacobian ranks between 1 and 4.

Abstract

We prove that for any number field $K$ and any fixed genus $g \geq 2$, there are infinitely many non-isomorphic hyperelliptic curves of genus $g$ over $K$ whose Jacobians have rank over $K$ equal to each of 0, 1, or 2. As an example of our method, over $\mathbb{Q}$, we prove that there exist infinitely many non-isomorphic hyperelliptic curves of genus two, whose Jacobians have rank equal to a fixed number between $1$ and $11$, genus three and four curves with rank between $1$ and $4$, and genus five and six with rank between $1$ and $3$.