Galois representations ramified at one prime via relative deformation theory
Abstract
Let p be an odd prime and let G/ℤ be a split connected reductive group with dim Z(G)≤1. Assume that a split maximal torus of G admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of Lie(Gder) modulo p. We construct infinitely many continuous representations ρ:G{p}⟶G(ℤp) which are unramified at every finite prime different from p and whose images contain a principal congruence subgroup of Gder(ℤp). When the center has dimension one, the representations may be chosen to have open image in G(ℤp). The construction continues the author's earlier work on GLn-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For G=GLn all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd p and every n>1, infinitely many representations G{p}→GLn(ℤp) with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction p≥7 in that construction.
Related
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- Irreducible components of crystabelline deformation rings: the wildly ramified case
- Non-admissible smooth irreducible representations of GL2(F) for totally ramified extensions F/ℚp
- Ramified tori and Gelfand--Kirillov dimension of mod p representations
- Fine structure of even Galois deformation rings
BibTeX
@misc{ray2026galois,
title = {{Galois representations ramified at one prime via relative deformation theory}},
author = {Anwesh Ray},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03954},
eprint = {2609.03954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03954},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Galois representations ramified at one prime via relative deformation theory
AU - Anwesh Ray
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03954
UR - https://arxiv.org/abs/2609.03954
AB - Let p be an odd prime and let G/ℤ be a split connected reductive group with dim Z(G)≤1. Assume that a split maximal torus of G admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction of Lie(G^der) modulo p. We construct infinitely many continuous representations ρ:G_{p}⟶G(ℤ_p) which are unramified at every finite prime different from p and whose images contain a principal congruence subgroup of G^der(ℤ_p). When the center has dimension one, the representations may be chosen to have open image in G(ℤ_p). The construction continues the author's earlier work on GL_n-valued representations ramified at one prime, but replaces the residual unobstructedness used there by a relative lifting argument. For G=GL_n all the required root-theoretic conditions are automatic for every odd prime. We therefore obtain, for every odd p and every n>1, infinitely many representations G_{p}→GL_n(ℤ_p) with open image. In particular, this removes the weak Vandiver-type hypothesis occurring in the author's previous work, as well as the restriction p≥7 in that construction.
ER -