| Summary: | 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
${\overline {{D}}}$
.
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