Mod ℓ gamma factors and a converse theorem for finite general linear groups

2023/07/14 by Jacksyn Bakeberg, Mathilde Gerbelli-Gauthier, Bakeberg, Jacksyn +9
Mathematics · #Advanced Algebra and Geometry #Finite Group Theory Research #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2307.07593

Abstract

The local converse theorem for Rankin-Selberg gamma factors of GL₂(\mathbbF_q) proved by Piatetski-Shapiro over ℂ no longer holds after reduction modulo ℓ ≠ p. To remedy this, we construct new GL_n × GL_m gamma factors valued in arbitrary ℤ[1/p, ζ_p]-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a GL_n × GL_n-1 converse theorem for irreducible cuspidal representations. In the GL₂ × GL₁ case, we define an alternative "new" gamma factor, which takes values in k and satisfies a converse theorem that matches the converse theorem in characteristic 0.

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