# Restrictions on endomorphism rings of jacobians and their minimal fields of definition

## TL;DR

This paper investigates how the structure of Galois groups, especially those with large prime order elements, influences the endomorphism rings and minimal fields of definition of Jacobians, extending previous results on their restrictions.

## Contribution

It provides a partial converse to existing theorems by analyzing Jacobians with Galois groups containing large prime order elements, broadening understanding of endomorphism restrictions.

## Key findings

- Galois groups with large prime order elements impose specific restrictions on Jacobian endomorphism rings.
- Partial converse to Guralnick and Kedlaya's results on endomorphism fields.
- Extensions of Zarhin's work to cases with less restrictive Galois group conditions.

## Abstract

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians $J$ for which the Galois group associated to their 2-torsion is insoluble and 'large' (relative to the dimension of $J$). In this paper we examine what happens when this Galois group merely contains an element of 'large' prime order. In doing so we obtain a partial converse to a result by Guralnick and Kedlaya on the endomorphism field.