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Title: The essential dimension of congruence covers
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.  more » « less
Award ID(s):
1944862 1811846
NSF-PAR ID:
10330092
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Compositio Mathematica
Volume:
157
Issue:
11
ISSN:
0010-437X
Page Range / eLocation ID:
2407 to 2432
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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