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Geometric Properties of Versal Deformation Rings and Universal Pseudodeformation Rings
Consider the absolute Galois group $G_K$ of an extension $K$ of $\mathbb{Q}_p$ finite degree $d$, and a finite field $\mathbb{F}$ of prime characteristic $p$. Following Mazur [Maz89], we define the versal deformation ring $R_{\overline{\rho}}^\psi$ with fixed determinant of a Galois representation $\overline{\rho}: G_K\to GL_n(\mathbb{F})$. Then for $n=2$ and $p>3$ our first main result states that $R_{\overline{\rho}}^\psi$ is an integral domain so that the associated versal deformation space $\mathfrak{X}(\overline{\rho})$ is irreducible. For this, we use the explicit relations of $R_{\overline{\rho}}^\psi$ computed in [Boe00] and a commutative algebra criterion. We deduce from [Nak13] that for $n=2$ and any $K$ the benign crystalline points are Zariski dense in $\mathfrak{X}({\overline{\rho}})$. This is expected to be useful for the surjectivity of the $p$-adic local Langlands correspondence. Furthermore, for arbitrary $n$ and $p$ we show that the refined quadratic parts of the relations of $R_{\overline{\rho}}^\psi$ can be obtained cohomologically from a cup product and a Bockstein homomorphism if a certain lift of $\overline{\rho}$ exists. Following Chenevier [Che14], we construct the universal pseudodeformation ring $R_{\overline{D}}^{univ}$ of an $n$-dimensional pseudorepresentation $\overline{D}: \mathbb{F}[G_K]\to\mathbb{F}$. Motivated by the result [Che11] on the equidimensionality of the generic fiber of the universal pseudorepresentation ring in characteristic $0$, our second main result says that the special fiber $\overline{R}_{\overline{D}}^{univ}$ of $R_{\overline{D}}^{univ}$ is equidimensional of dimension $dn^2+1$ if $p>n$ or if $K$ does not contain a primitive $p^{th}$ root of unity $\zeta_p$. In the latter case, if either $n>2$ or $n=2$ and $d>1$ we prove that the regular locus of $\Spec\overline{R}_{\overline{D}}^{univ}$ consists of certain irreducible pseudodeformations and that $\overline{R}_{\overline{D}}^{univ}$ satisfies Serre's condition $R_2$.
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510 Mathematik
510 Mathematics