Marco Boggi

Let $C$ be a very general complex smooth projective algebraic curve endowed with a group of automorphisms $G$ such that the quotient $C/G$ has genus at least 3. Then, the algebra of $\mathbb{Q}$-endomorphisms of the Jacobian $J(C)$ of $C$ is naturally isomorphic to the group algebra $\mathbb{Q}G$. Via Hodge theory this result has applications to the theory of virtual linear representations of the mapping class group. In the first part of the talk, I will explain some of these applications and why they are important. In the second part, I will try to give an idea of the proof. This talk is based on joint work with Eduard Looijenga (cf. arXiv:1811.09741).