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The essential dimension of congruence covers

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  27 October 2021

Benson Farb ,
Mark Kisin  and
Jesse Wolfson
Benson Farb
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL60637, USAfarb@math.uchicago.edu
Mark Kisin
Affiliation:
Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford Street, Cambridge, MA02138, USAkisin@math.harvard.edu
Jesse Wolfson
Affiliation:
Department of Mathematics, University of California–Irvine, Rowland Hall 340H, Irvine, CA92697, USAwolfson@uci.edu
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Abstract

Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

Keywords

Shimura varietiesessential dimensionfinite flat group schemes

MSC classification

Primary: 14G35: Modular and Shimura varieties
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 11 , November 2021 , pp. 2407 - 2432
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The authors are partially supported by NSF grants DMS-1811772 (BF), DMS-1601054 (MK) and DMS-1811846 (JW).

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