The essential dimension of congruence covers

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.

Automated summary

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.