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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.
