Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields

Abstract

Let K be a finite extension of the p-adic field ${\mathbb {Q}}_p$ of degree d, let ${{\mathbb {F}}\,\!{}}$ be a finite field of characteristic p and let ${\overline {{D}}}$ be an n-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of K over the field ${{\mathbb {F}}\,\!{}}$ . For the universal mod p pseudodeformation ring ${\overline {R}{{\phantom {\overline {\overline m}}}}^{\operatorname {univ}}_{{{\overline {{D}}}}}}$ of ${\overline {{D}}}$ , we prove the following: The ring $\overline R_{{\overline {{D}}}}^{\mathrm {ps}}$ is equidimensional of dimension $dn^2+1$ . Its reduced quotient ${\overline {R}{{\phantom {\overline {\overline m}}}}^{\operatorname {univ}}_{{{\overline {{D}}},{\operatorname {red}}}}}$ contains a dense open subset of regular points x whose associated pseudocharacter ${D}_x$ is absolutely irreducible and nonspecial in a certain technical sense that we shall define. Moreover, we will characterize in most cases when K does not contain a p-th root of unity the singular locus of ${\mathrm {Spec}}\ {\overline {R}{{\phantom {\overline {\overline m}}}}^{\operatorname {univ}}_{{{\overline {{D}}}}}}$ . Similar results were proved by Chenevier for the generic fiber of the universal pseudodeformation ring ${R{{\phantom {\overline {m}}}}^{\operatorname {univ}}_{{{\overline {D}}}}}$ of ${\overline {{D}}}$ .