On the dimension conjecture for global Galois deformation rings in characteristic two
Abstract
Let K be a number field, and let S be a finite set of primes of K containing all primes above $2$ and all archimedean primes. Let GK,S be the Galois group of the maximal algebraic extension of K unramified outside S. We prove two refined lower bounds for the Krull dimension of unrestricted global Galois deformation rings of continuous absolutely irreducible representations of GK,S over finite fields of characteristic $2$. As applications, we construct the first counterexamples to the dimension conjecture in every dimension n≥ 2 over every number field with a real prime. Over every totally real number field, these constructions also yield the first examples of such universal deformation rings that fail to be complete intersections. Furthermore, we propose a modified dimension conjecture for K=ℚ and n=2.
Related
- Fine structure of even Galois deformation rings
- Galois deformation rings and modularity in the residually reducible case
- Anabelian presentation of the motivic Galois group in characteristic zero
- Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
- Galois representations ramified at one prime via relative deformation theory
BibTeX
@misc{luo2026dimension,
title = {{On the dimension conjecture for global Galois deformation rings in characteristic two}},
author = {Yufan Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21298},
eprint = {2610.21298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21298},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On the dimension conjecture for global Galois deformation rings in characteristic two
AU - Yufan Luo
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21298
UR - https://arxiv.org/abs/2610.21298
AB - Let K be a number field, and let S be a finite set of primes of K containing all primes above $2$ and all archimedean primes. Let G_K,S be the Galois group of the maximal algebraic extension of K unramified outside S. We prove two refined lower bounds for the Krull dimension of unrestricted global Galois deformation rings of continuous absolutely irreducible representations of G_K,S over finite fields of characteristic $2$. As applications, we construct the first counterexamples to the dimension conjecture in every dimension n≥ 2 over every number field with a real prime. Over every totally real number field, these constructions also yield the first examples of such universal deformation rings that fail to be complete intersections. Furthermore, we propose a modified dimension conjecture for K=ℚ and n=2.
ER -