$p$-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 $G$ be a Barsotti-Tate over $k$. We denote by $\bS$ the «algebraic» local moduli in characteristic $p$ of $G$, by $\bG$ the universal deformation of $G$ over $\bS$, and by $\bU\subset\bS$ the ordinary locus of $\bG$. The étale part of $\bG$ over $\bU$ gives rise to a monodromy representation $\rho_{\bG}$ of the fundamental group of $\bU$ on the Tate module of $\bG$. Motivated by a famous theorem of Igusa, we prove in this article that $\rho_{\bG}$ 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 $k$.

2000 Mathematics Subject Classification: 13D10, 14L05, 14H30, 14B12, 14D15, 14L15

Keywords and Phrases: Barsotti-Tate groups ($p$-divisible groups), $p$-adic monodromy representation, universal deformation, Hasse-Witt maps.

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