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.
Keywords
Cite
@article{arxiv.1811.09741,
title = {Curves with prescribed symmetry and associated representations of mapping class groups},
author = {Marco Boggi and Eduard Looijenga},
journal= {arXiv preprint arXiv:1811.09741},
year = {2022}
}
Comments
Aaron Landesman and Daniel Litt pointed out to us that the proofs of Lemma 1.6 and Lemma 1.8 are incomplete. While we believe that the proof of Lemma 1.8 can be completed, the gap in the proof of Lemma 1.6 is of a more fundamental nature and leads us to withdraw the paper, as its main result (Theorem 1.1) depends on this