Spectral Reciprocity for the first moment of triple product L-functions and applications

2024/12/27 by Xinchen Miao, Miao, Xinchen
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2501.10418

openalex publication_date 2024/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a number field with adele ring \mathbbA_F, π₁, π₂ be two fixed unitary automorphic representations of PGL₂(\mathbbA_F) with finite coprime analytic conductor \mathfraku and \mathfrakv, \mathfrakq,\mathfrakl be two coprime integral ideals with (\mathfrakq \mathfrakl, \mathfraku \mathfrakv)=1. Following [Zac20], we estimate the first moment of L((1)/(2), π⊗ π₁ ⊗ π₂) twisted by the Hecke eigenvalues λ_π(\mathfrakl), where π runs over unitary automorphic representations of finite conductor dividing \mathfraku\mathfrakv\mathfrakq. By applying the triple product integrals, spectral decomposition and Plancherel formula, we get a reciprocity formula links the twisted first moment of triple product L-functions to the spectral expansion of certain triple product periods over automorphic representations of finite conductor dividing \mathfrakl. As application, we study the subconvexity problem for the triple product L-function in the level aspect and give a subconvex bound for L((1)/(2), π⊗ π₁ ⊗ π₂) in terms of the norm of \mathfrakq.

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