Irreducibility of versal deformation rings in the \((p,p)\)-case for 2-dimensional representations
Let \(K\) be a finite extension of \({\mathbb Q}_p\) and \(G_K\) be its absolute Galois group. Let further \({\overline\rho}: G_{K} \rightarrow GL_{n}({\mathbb F}) \) be a continuous residual representation for \({\mathbb F}\) a finite field of characteristic \(p\). Denote by \(W ({\mathbb F})\) the ring of Witt vectors of \({\mathbb F}\) and by \(\mathcal O\) the ring of integers of a finite totally ramified extension of \(W ({\mathbb F})[1/p].\) Denote also by \({\mathfrak m}_{\mathcal O}\) its maximal ideal and by \({\overline{\omega}}_{\mathcal O}\) a uniformizer. For \({\overline\rho},\) there exists the deformation functor from the category \({\widehat{Ar_{\mathcal O}}}\) of complete Noetherian local \({\mathcal O}\)-algebras \((R,{\mathfrak m}_R)\) to the category of sets defined by \textit{Ph. Cassou-Noguès} and \textit{M. J. Taylor} (cf. [Prog. Math. 75, 35--64 (1988; Zbl 0714.11078)]) \[ D_{\overline{\rho}}(R):=\{{\rho}: G_{K} \rightarrow GL_{n}(R) \mid \rho \operatorname{mod} {\mathfrak m}_R ={\overline{\rho}}{\text{ and }}\rho {\text{ is a cont. representation}}\}/{\sim} \] where \({\rho}\sim {\rho}'\) if there exists \(A\in {\text{ker}} \,\{ GL_{n}(R) \rightarrow GL_{n}({\mathbb F})\}\) such that \({\rho}' = A{\rho}A^{-1}.\) A deformation of \({\overline{\rho}}\) is an equivalence class \([{\rho}] \) of \({\rho}\) under \(\sim\). The authors study the versal deformation space \({\mathfrak X}({\overline{\rho}})\) of \({\overline{\rho}}.\) In particular the question of its irreducibility. They show that the answer is affirmative for \(n = 2\) and \(p > 2\) and that the crystalline points are Zariski dense in the versal deformation space \({\mathfrak X}({\overline{\rho}}).\)
scientific article; zbMATH DE number 6493620
scientific article