Curves with prescribed symmetry and associated representations of mapping class groups

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 QG to the algebra of 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 Q-irreducibly in a G-isogeny space of H^1(C; Q)and with image often a Q-almost simple group.