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arXiv · 1901.09013

The Essential Dimension of Congruence Covers

Benson Farb·Mark Kisin·Jesse Wolfson
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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 Kronecker and 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.

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Benson Farb, Mark Kisin, Jesse Wolfson. 2019-01-25. The Essential Dimension of Congruence Covers. https://doi.org/10.1112/s0010437x21007594

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