\(p\)-adic monodromy of the universal deformation of a HW-cyclic Barsotti-Tate group (Q845265)

Summary: 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 \(k\).