Algebraicity of adjoint L-functions for quasi-split groups

2025/09/28 by Shih-Yu Chen, Chen, Shih-Yu
Mathematics · #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.23940

openalex publication_date 2025/09/28 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

For a globally generic cuspidal automorphic representation \mathitΠ of a quasi-split reductive group G over \mathbb Q, E. Lapid and Z. Mao proposed a conjecture on the decomposition of the global Whittaker functionals on \mathitΠ into products of an adjoint L-value of \mathitΠ and the local Whittaker functionals. In this paper, we consider the algebraic aspect of the Lapid-Mao conjecture. More precisely, when \mathitΠ is C-algebraic, we show that the algebraicity of the adjoint L-value can be expressed in terms of the Petersson norm of Whittaker-rational cusp forms in \mathitΠ, subject to the validity of the Lapid-Mao conjecture. For unitary similitude groups, we also establish an unconditional and more refined algebraicity result. Additionally, we give an explicit formula for the case G=\rm U(2,1).

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Related

• Holomorphy of adjoint L functions for quasisplit A2
• On a certain local identity for Lapid-Mao's conjecture and formal degree conjecture : even unitary group case
• On Ichino-Ikeda type formula of Whittaker periods for unitary groups
• Adjoint L-functions, congruence ideals, and Selmer groups over GL_n
• Basic functions and unramified local L-factors for split groups