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

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 $k$.