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The essential dimension of congruence covers
Published online by Cambridge University Press: 27 October 2021
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
Footnotes
The authors are partially supported by NSF grants DMS-1811772 (BF), DMS-1601054 (MK) and DMS-1811846 (JW).
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