Fixed points, local monodromy, and incompressibility of congruence covers

Brosnan, Patrick; Fakhruddin, Najmuddin

doi:10.1090/jag/800

Authors: Patrick Brosnan and Najmuddin Fakhruddin
Journal: J. Algebraic Geom. 33 (2024), 295-346
DOI: https://doi.org/10.1090/jag/800
Published electronically: June 5, 2023
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Abstract | References | Additional Information

Abstract: We prove a fixed point theorem for the action of certain local monodromy groups on étale covers and use it to deduce lower bounds on essential dimension. In particular, we give more geometric proofs of some of the results of a paper of Farb, Kisin and Wolfson, which uses arithmetic methods to prove incompressibility results for Shimura varieties and moduli spaces of curves. Our method allows us to prove new results for exceptional groups, applies also to the reduction modulo good primes of congruence covers of Shimura varieties and moduli spaces of curves, and also to certain “quantum” covers of moduli spaces of curves arising from a certain TQFT.

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Additional Information

Patrick Brosnan
Affiliation: Department of Mathematics, 1301 Mathematics Building, University of Maryland, College Park, Maryland 20742-4015
MR Author ID: 707674
Email: pbrosnan@umd.edu

Najmuddin Fakhruddin
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
MR Author ID: 604412
Email: naf@math.tifr.res.in

Received by editor(s): September 19, 2021
Published electronically: June 5, 2023
Additional Notes: The first author was supported by the Simons Foundation for a Collaboration Grant. The second author was supported by the DAE, Government of India, under project no. RTI4001.
Article copyright: © Copyright 2023 University Press, Inc.