Irreducible components of crystabelline deformation rings: the wildly ramified case
Abstract
Let p≥11 be a prime, and let ρ̅:Gal(QQ̅p/QQp)→GL2(FF̅p) be a reducible and very generic representation. In this paper, we use ghost series \cite{Lghost2} to study the irreducible components of Kisin's potentially crystalline deformation rings of ρ̅. For wildly ramified principal series inertial types and arbitrary regular Hodge--Tate weights, we distinguish the components by the crystabelline slopes on their generic fibers and determine the total number of irreducible components. Combining this with an explicit formula for the total Hilbert--Samuel multiplicity of the special fiber obtained from the Breuil--Mézard conjecture, we compute the Hilbert--Samuel multiplicity of the special fiber of each component. Every such multiplicity is one when ρ̅ is non-split, and is either one or two when ρ̅ is split, according to whether the generic fiber of the corresponding component parametrizes reducible or irreducible Galois representations.
Related
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BibTeX
@misc{an2026irreducible,
title = {{Irreducible components of crystabelline deformation rings: the wildly ramified case}},
author = {Jiawei An},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16225},
eprint = {2610.16225},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16225},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Irreducible components of crystabelline deformation rings: the wildly ramified case
AU - Jiawei An
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16225
UR - https://arxiv.org/abs/2610.16225
AB - Let p≥11 be a prime, and let ρ̅:Gal(QQ̅_p/QQ_p)→GL_2(FF̅_p) be a reducible and very generic representation. In this paper, we use ghost series \cite{Lghost2} to study the irreducible components of Kisin's potentially crystalline deformation rings of ρ̅. For wildly ramified principal series inertial types and arbitrary regular Hodge--Tate weights, we distinguish the components by the crystabelline slopes on their generic fibers and determine the total number of irreducible components. Combining this with an explicit formula for the total Hilbert--Samuel multiplicity of the special fiber obtained from the Breuil--Mézard conjecture, we compute the Hilbert--Samuel multiplicity of the special fiber of each component. Every such multiplicity is one when ρ̅ is non-split, and is either one or two when ρ̅ is split, according to whether the generic fiber of the corresponding component parametrizes reducible or irreducible Galois representations.
ER -