Monodromy and irreducibility of leaves

Abstract

Let $p$ be a prime number, let $g$ and $d$ be positive integers, and let ${\cal A}_{g,d}$ be the moduli space over an algebraic closure of the prime field $\mathbb{F}_p$ which classifies $g$-dimensional abelian varieties with a polarization of degree $d^2$. We study stratifications and foliations of these spaces.

It is known that Newton polyton strata connected with supersingular abelian varieties are reducible (for large $p$). In this paper we show:

• (a)  every non-supersingular Newton polygon stratum in ${\cal A}_{g,1}$ is irreducible,
• (b)  every non-supersingular leaf in ${\cal A}_{g,d}$ is (geometrically) irreducible and
• (c)  the $p$-adic monodromy for every non-supersingular leaf in ${\cal A}_{g,d,n}$ maximal. Here
  $n$ indicates any prime-to-$p$ level structure.

The proofs are a mixture of geometry, number theory, and the theory of linear groups. We use: if a reduced subscheme $Z$ of ${\cal A}_{g,d}$ is stable under all prime-to-$p$ Hecke correspondences and these correspondences operative transitively on the set $\Pi(Z)$ of geometrically irreducible compoments of $Z$, then $Z$ is geometrically irreducible.

A result of deformation to $a\le 1$ and the way Newton polygon strata fit together proves (a).

Then we leverage (a) with the notion of hypersymmetric points and strong approximation to get (b).

We finish by proving (c).

Note that in (a) we work with polarizations (and generalizations to non-principal polarizations fail). However over leaves, as in (b) and (c), the degree of the polarization is arbitrary.