$p$-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, andG be a Barsotti-Tate overk. We denote by S the algebraic local moduli in characteristic p of G, by G the universal deformation of G over S, and by U S the ordinary locus of G. The etale part of G over U gives rise to a monodromy representation G 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 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.