Characterizing the mod-ℓ local Langlands correspondence by nilpotent gamma factors

2019/05/31 by Gilbert Moss, Moss, Gilbert
Mathematics · #11S37 #11S40 #20C20 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1905.13487

openalex publication_date 2019/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a p-adic field and choose k an algebraic closure of \mathbbF_ℓ, with ℓ different from p. We define ``nilpotent lifts'' of irreducible generic k-representations of GL_n(F), which take coefficients in Artin local k-algebras. We show that an irreducible generic ℓ-modular representation π of GL_n(F) is uniquely determined by its collection of Rankin--Selberg gamma factors γ(π× \widetildeτ,X,ψ) as \widetildeτ varies over nilpotent lifts of irreducible generic k-representations τ of GL_t(F) for t=1,…, \lfloor (n)/(2)\rfloor. This gives a characterization of the mod-ℓ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

Citations

• Deligne--Langlands gamma factors in families

Related

• Rankin-Selberg local factors modulo ℓ
• On the local Langlands correspondence
• Converse theorems and the local Langlands correspondence in families
• The ℓ-modular local Langlands correspondence and local factors
• On the dependence of the local Rankin-Selberg gamma factors of \textrmSp_2n× \textrmGL_m on ψ