Galois deformation rings and modularity in the residually reducible case
Abstract
We prove some residually reducible p-adic Galois representations of number fields arise from modular forms. We study the universal deformation ring arising from deformations satisfying the Fontaine-Laffaille condition at primes over p. Under certain conditions, we establish the reduced universal deformation ring is a discrete valuation ring. The method uses pseudocharacters and certain bounds on Selmer groups, where the ideal of reducibility as defined by Bellaïche and Chenevier is shown to be maximal and principal. A self-dual assumption is not needed in our argument. The main result on deformations applies to n-dimensional representations. In applications, a congruence between Hermitian modular forms due to Klosin is used with our results to prove modularity of some 4-dimensional p-adic representations of the imaginary quadratic field Q(i).
Versions
Related
- On the modularity of residually reducible Galois representations of rank 2
- Some instances of modularity for residually reducible representations in the mixed case
- Fine structure of even Galois deformation rings
- On the dimension conjecture for global Galois deformation rings in characteristic two
- Galois representations and modularity for twisted moments of Kloosterman sums
BibTeX
@misc{akers2026galois,
title = {{Galois deformation rings and modularity in the residually reducible case}},
author = {Geoffrey Akers},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21156},
eprint = {2609.21156},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21156},
}
RIS
TY - UNPB
TI - Galois deformation rings and modularity in the residually reducible case
AU - Geoffrey Akers
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21156
UR - https://arxiv.org/abs/2609.21156
AB - We prove some residually reducible p-adic Galois representations of number fields arise from modular forms. We study the universal deformation ring arising from deformations satisfying the Fontaine-Laffaille condition at primes over p. Under certain conditions, we establish the reduced universal deformation ring is a discrete valuation ring. The method uses pseudocharacters and certain bounds on Selmer groups, where the ideal of reducibility as defined by Bellaïche and Chenevier is shown to be maximal and principal. A self-dual assumption is not needed in our argument. The main result on deformations applies to n-dimensional representations. In applications, a congruence between Hermitian modular forms due to Klosin is used with our results to prove modularity of some 4-dimensional p-adic representations of the imaginary quadratic field Q(i).
ER -