Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

TL;DR

This paper demonstrates that in certain families of hyperelliptic curves, a density-one subset has Jacobians with maximal Galois action on torsion points, supported by explicit examples for genus 2 and 3.

Contribution

It applies a general theorem to hyperelliptic curve families, showing most members have maximal Galois representations on Jacobian torsion points and provides explicit examples.

Findings

Density-one subset with maximal Galois image in each family.

Explicit examples of hyperelliptic Jacobians with maximal Galois action.

Application of a general theorem to specific hyperelliptic cases.

Abstract

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.