Mathematische Annalen | 2021
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 $${{\\mathbb {Q}}}G$$\n \n Q\n G\n \n to the algebra of $${{\\mathbb {Q}}}$$\n Q\n -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 often acts $${{\\mathbb {Q}}}$$\n Q\n -irreducibly in a G-isogeny space of $$H^1(C; {{\\mathbb {Q}}})$$\n \n \n H\n 1\n \n \n (\n C\n Íž\n Q\n )\n \n \n and with image a $${{\\mathbb {Q}}}$$\n Q\n -almost simple group.