Reduction of Weil-Deligne Representations
Abstract
Let p and ℓ be distinct odd primes. For a finite extension F/ℚp, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of GLn(F) and n-dimensional, Φ-semisimple Weil--Deligne representations of the Weil group WF. Given two $2$-dimensional, semisimple, continuous representations ρ1, ρ2 of WF with images in GL2(𝒪K), where K/ℚℓ is a finite extension with maximal ideal λ⊂ 𝒪K, a natural question to ask is when their mod λ reductions ρ̅1 and ρ̅2 are isomorphic. In this paper, we give a complete characterization of when the reductions are isomorphic. For this, we first give a description of when the reductions of these representations are decomposable or irreducible, utilizing the correspondence between continuous ℓ-adic representations of WF and ℓ-adic Weil--Deligne representations. We conclude with some examples which arise in the modular method for solving generalized Fermat equations.
Related
- Siegel-Weil type identities for general linear groups related to parabolic induction from two Speh representations
- Hodge--Deligne polynomials and Sn-representations on the cohomology of ℳ̅2,n
- Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law
- Reduction modulo p of crystalline Galois representations via μ_p-equivariance
- The Deligne--Drinfeld conjecture after OpenAI
BibTeX
@misc{chen2026reduction,
title = {{Reduction of Weil-Deligne Representations}},
author = {Imin Chen and Deniz Suozer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36467},
eprint = {2609.36467},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36467},
}
RIS
TY - UNPB
TI - Reduction of Weil-Deligne Representations
AU - Imin Chen
AU - Deniz Suozer
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36467
UR - https://arxiv.org/abs/2609.36467
AB - Let p and ℓ be distinct odd primes. For a finite extension F/ℚ_p, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of GL_n(F) and n-dimensional, Φ-semisimple Weil--Deligne representations of the Weil group W_F. Given two $2$-dimensional, semisimple, continuous representations ρ_1, ρ_2 of W_F with images in GL_2(𝒪_K), where K/ℚ_ℓ is a finite extension with maximal ideal λ⊂ 𝒪_K, a natural question to ask is when their mod λ reductions ρ̅_1 and ρ̅_2 are isomorphic. In this paper, we give a complete characterization of when the reductions are isomorphic. For this, we first give a description of when the reductions of these representations are decomposable or irreducible, utilizing the correspondence between continuous ℓ-adic representations of W_F and ℓ-adic Weil--Deligne representations. We conclude with some examples which arise in the modular method for solving generalized Fermat equations.
ER -