Journal
COMPOSITIO MATHEMATICA
Volume 157, Issue 11, Pages 2407-2432Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1112/S0010437X21007594
Keywords
Shimura varieties; essential dimension; finite flat group schemes
Categories
Funding
- NSF [DMS-1811772, DMS-1601054, DMS-1811846]
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This study answers a question by calculating the essential dimension and p-dimension of congruence covers of the moduli space of principally polarized abelian varieties. The results are then applied to compute the essential p-dimension of congruence covers of curves, hyperelliptic loci, and locally symmetric varieties. These findings provide the first examples of nontrivial lower bounds on the essential dimension of proper algebraic varieties.
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.
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