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 Kronecker and Klein asks: What is minimal $d$ such that, after rational change of variables, can be written as variables? Using techniques from deformation theory $p$-divisible groups finite flat group schemes, we answer this computing essential dimension $p$-dimension congruence covers moduli space principally polarized varieties. We apply result compute genus $g$ curves, well its hyperelliptic locus, certain locally symmetric