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

Let K be a finite extension of the p-adic field Q𝑝 of degree d, let F be a finite field of characteristic p and let 𝐷 be an n-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of K over the field F. For the universal mod p pseudodeformation ring 𝑅 univ 𝐷 of 𝐷, we prove the following: The ring 𝑅ps 𝐷 is equidimensional of dimension 𝑑𝑛2 +1. Its reduced quotient 𝑅 univ 𝐷, red contains a dense open subset of regular points x whose associated pseudocharacter 𝐷𝑥 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 Spec 𝑅 univ 𝐷. Similar results were proved by Chenevier for the generic fiber of the universal pseudodeformation ring 𝑅univ 𝐷 of 𝐷. [ABSTRACT FROM AUTHOR]