$p$-adic monodromy of the universal deformation of a HW-cyclic Barsotti–Tate group

Abstract

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