p-adic Monodromy of the Universal Deformation of a HW-cyclic Barsotti-Tate Group

TL;DR

This paper proves that the monodromy representation associated with the universal deformation of certain connected, HW-cyclic Barsotti-Tate groups over an algebraically closed field is surjective, extending classical results in p-adic geometry.

Contribution

It establishes the surjectivity of the monodromy representation for HW-cyclic Barsotti-Tate groups, generalizing Igusa's theorem to a broader class of p-divisible groups.

Findings

Monodromy representation is surjective for connected HW-cyclic groups.

The HW-cyclic condition is equivalent to the a-number being 1.

Results apply to all connected one-dimensional Barsotti-Tate groups.

Abstract

Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the "algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $\rho$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $\rho$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivalent to that Oort's $a$-number of $G_0$ equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$.