---
abstract: 'Let $C$ be a complex smooth projective algebraic curve endowed with an action of a finite group $G$ such that the quotient curve has genus at least $3$. We prove that if the $G$-curve $C$ is very general for these properties, then the natural map from the group algebra ${{\mathbb Q}}G$ to the algebra of ${{\mathbb Q}}$-endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation acts ${{\mathbb Q}}$-irreducibly in a $G$-isogeny space of $H^1(C; {{\mathbb Q}})$ and with image often a ${{\mathbb Q}}$-almost simple group.'
address:
- 'Departamento de Matemática, UFMG, Belo Horizonte - MG (Brasil)'
- 'Yau Mathematical Sciences Center, Tsinghua University Beijing (China) and Mathematisch Instituut, Universiteit Utrecht (Nederland)'
author:
- Marco Boggi
- Eduard Looijenga
title: Curves with prescribed symmetry and associated representations of mapping class groups
---

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Introduction {#introduction .unnumbered}
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A classical theorem of Hurwitz (1886) asserts that for a *very general* complex projective smooth curve $C$ of genus $\ge 2$, the endomorphism ring of its Jacobian $J(C)$ is as small 