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 ${\mathbb {Q}}_p$ 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 f...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000828/type/journal_article |
| _version_ | 1797614840686051328 |
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| author | Gebhard Böckle Ann-Kristin Juschka |
| author_facet | Gebhard Böckle Ann-Kristin Juschka |
| author_sort | Gebhard Böckle |
| collection | DOAJ |
| description | 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
${\overline {{D}}}$
. |
| first_indexed | 2024-03-11T07:17:58Z |
| format | Article |
| id | doaj.art-c901560ab4944fe19d9ffa7c58971382 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-11T07:17:58Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-c901560ab4944fe19d9ffa7c589713822023-11-17T08:11:57ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.82Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fieldsGebhard Böckle0https://orcid.org/0000-0003-1758-1537Ann-Kristin Juschka1Universität Heidelberg, IWR and Institute of Mathematics, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany; E-mail: ,Universität Heidelberg, IWR and Institute of Mathematics, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany; E-mail: ,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 ${\overline {{D}}}$ .https://www.cambridge.org/core/product/identifier/S2050509423000828/type/journal_article11F8011F8511F70 |
| spellingShingle | Gebhard Böckle Ann-Kristin Juschka Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields Forum of Mathematics, Sigma 11F80 11F85 11F70 |
| title | Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields |
| title_full | Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields |
| title_fullStr | Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields |
| title_full_unstemmed | Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields |
| title_short | Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields |
| title_sort | equidimensionality of universal pseudodeformation rings in characteristic p for absolute galois groups of p adic fields |
| topic | 11F80 11F85 11F70 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000828/type/journal_article |
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