Periods and Reciprocity II

2019/12/03 by Zacharias, Raphaël · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.01512

Abstract

Let F be a number field and \mathfrakq,\mathfrakl two coprime integral ideals with \mathfrakq squarefree and π₁,π₂ two fixed unitary automorphic representations of PGL₂(\mathbbA_F) unramified at all finite places. In this paper, we use regularized integrals to obtain a formula that links the first moment of L(π⊗π₁⊗π₂,\tfrac12) twisted by the Hecke eigenvalues λ_π(\mathfrakl), where π runs through unitary automorphic representations of PGL₂(\mathbbA_F) with conductor dividing \mathfrakq, with some spectral expansion of periods over representations of conductor dividing \mathfrakl. In the special case where π₁=π₂=σ, this formula becomes a reciprocity relation between moments of L-functions. As applications, we obtain a subconvex estimate in the level aspect for the central value of the triple product L(π⊗π₁⊗π₂,\tfrac12) and a simultaneous non-vanishing result for L(Sym²(σ)⊗ π,\tfrac12) and L(π,\tfrac12).

Cited by

• Stationary phase analysis for analytic newvectors and application to subconvexity problems

Related

• Periods and Reciprocity I
• Spectral Reciprocity for the first moment of triple product L-functions and applications
• Spectral Reciprocity and Hybrid Subconvexity Bound for triple product L-functions
• Spectral reciprocity for GL(n) and simultaneous non-vanishing of central L-values
• Spectral Reciprocity: A Fourier--Analytic Approach