The completed standard L-function of modular forms on G₂

2021/04/19 by Fatma Çiçek, Çiçek, Fatma, Giuliana P. Davidoff +10 · 1 voice
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT #math.RT

paper · pdf · doi:10.48550/arxiv.2104.09448

openalex publication_date 2021/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to provide a complete and refined study of the standard L-functions L(π,Std,s) for certain non-generic cuspidal automorphic representations π of G₂(\mathbbA). For a cuspidal automorphic representation π of G₂(\mathbbA) that corresponds to a modular form φ of level one and of even weight on G₂, we explicitly define the completed standard L-function, Λ(π,Std,s). Assuming that a certain Fourier coefficient of φ is nonzero, we prove the functional equation Λ(π,Std,s) = Λ(π,Std,1-s). Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.

Discussions

• 👋 Trajan was a participant in the first Rethinking Number Theory workshop and co-authored this paper: arxiv.org/abs/2104.09448 [bsky, 5 points, 1 comments]

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