On the connected components of moduli spaces of Kisin modules (Q420662)

Let \(K\) be a \(p\)-adic field, and let \(V_{\mathbb{F}}\) be a two-dimensional continuous representation of the absolute Galois group \(G_K\) over a finite field \(\mathbb{F}\) of characteristic \(p\). Let \(M_{\mathbb{F}}\) be a \(\phi\)-module corresponding to the Galois representation \(V_{\mathbb{F}}(-1)\), and \(\mathcal{GR}_{V_{\mathbb{F}}, 0}\) be the moduli space of Kisin modules in \(M_{\mathbb{F}}\) constructed in \textit{M. Kisin}'s paper [Ann. Math. (2) 170, No. 3, 1085--1180 (2009; Zbl 1201.14034)]. The author proves the followingNEWLINENEWLINETheorem. The non-ordinary locus of \(\mathcal{GR}_{V_{\mathbb{F}}, 0}^{\mathbf v}\), a closed subscheme of \(\mathcal{GR}_{V_{\mathbb{F}}, 0}\) determined by the condition that \(p\)-adic Hodge type is \({\mathbf v}=1\), is geometrically connected.NEWLINENEWLINEThe case when \(p>2\) was already proved by the author in a previous paper [Am. J. Math. 132, No. 5, 1189--1204 (2010; Zbl 1205.14025)]. This paper completes the proof of the theorem by proving the case when \(p=2\).