RETRACTED ARTICLE: Curves with prescribed symmetry and associated representations of mapping class groups

Marco Boggi,Eduard Looijenga

doi:10.1007/s00208-021-02234-2

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}G$$\\end{document} to the algebra of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}-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 Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}-irreducibly in a G-isogeny space of H1(C;Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1(C; {{\\mathbb {Q}}})$$\\end{document} and with image a Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}-almost simple group.

Highlights

A classical theorem of Hurwitz (1886) asserts that for a very general complex projective smooth curve C of genus ≥ 2, the endomorphism ring of its Jacobian J (C) is as small as possible, namely Z
Ciliberto, van der Geer and Teixidor i Bigas [5] studied the locus of curves for which the endomorphism ring of the Jacobian is strictly larger than Z, while Zarhin addressed this problem over other base fields and considered in [24,25] curves with a specified automorphism of a particular type other than a hyperelliptic involution: he proved that for a general such curve the endomorphism ring of its Jacobian is as small as it could possibly be
Mod(S)G and Mod(SG) have in common a subgroup of finite index so that the natural representation ρG : Mod(S)G → Sp(H 1(S, Q))G, where we denote by Sp(H 1(S, Q))G the centralizer of G in the symplectic group Sp(H 1(S, Q)), can be regarded as a virtual linear representation of the mapping class group Mod(SG)

Summary

Introduction

A classical theorem of Hurwitz (1886) asserts that for a very general complex projective smooth curve C of genus ≥ 2, the endomorphism ring of its Jacobian J (C) is as small as possible, namely Z. In the present paper we obtain a natural generalization of these results, at least over the base field C, by proving such a minimality property for curves endowed with an action of a given (but arbitrary) finite group. Mod(S)G and Mod(SG) have in common a subgroup of finite index so that the natural representation ρG : Mod(S)G → Sp(H 1(S, Q))G , where we denote by Sp(H 1(S, Q))G the centralizer of G in the symplectic group Sp(H 1(S, Q)), can be regarded as a virtual linear representation of the mapping class group Mod(SG). Let us describe this centralizer in more detail. We adhere to the tradition in algebraic geometry (where sheaf cohomology with supports takes the place of relative cohomology) to use the comma for denoting sheaf cohomology, e.g., Hr (X , OX )

Endomorphisms of Jacobians of G-curves

Symplectic G-modules and associated unitary groups

Virtual linear representations of the mapping class group