# The Essential Dimension of Congruence Covers

## TL;DR

This paper computes the minimal number of parameters needed to describe congruence covers of moduli spaces of abelian varieties and curves, using deformation theory and essential dimension concepts.

## Contribution

It introduces new methods to determine the essential dimension of congruence covers of moduli spaces, connecting deformation theory with algebraic geometry.

## Key findings

- Computed the essential dimension of congruence covers of moduli spaces
- Determined the $p$-dimension of these covers
- Applied results to moduli of curves and hyperelliptic loci

## 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 Kronecker and 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.