Products of hyperelliptic Jacobians with maximal Galois image

TL;DR

This paper investigates the Galois representations of products of hyperelliptic Jacobians, providing criteria for maximal Galois image and methods to construct Jacobians with disjoint division fields.

Contribution

It introduces a simple criterion for maximal Galois image of adelic representations and offers a novel method to construct Jacobian products with disjoint division fields.

Findings

Criterion for maximal Galois image established

Method for constructing Jacobians with disjoint division fields

Enhanced understanding of Galois representations of hyperelliptic Jacobians

Abstract

In this note we study the associated adelic representation of a product of hyperelliptic Jacobians. We give a simple criterion that assures that this representation has maximal Galois image in a certain sense. As an application, we provide a method of constructing products of Jacobians with division fields as disjoint as they can be.