Stationary phase analysis for analytic newvectors and application to subconvexity problems

2025/11/27 by Lideng Ye, Ye, Liyuan
Mathematics · #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2511.22644

Abstract

In this paper, we extend the results of Michel-Venkatesh and Hu-Michel-Nelson to establish an upper bound for triple product and Rankin-Selberg L-functions of the form L(π₁ ⊗ π₂ ⊗ π₃,(1)/(2))≪_π3,εC(π₁⊗π₂)^{(1)/(2) + ε} ( (C(π₁ ⊗ π₂))/(C(π₂ ⊗ π₂)))^-δ in the spectral aspect, allowing conductor dropping. In particular, we obtain a subconvexity bound when π₁⊗π₂ stays uniformly away from QUE-like case. The new ingredient is a stationary phase analysis of the analytic newvectors introduced by Jana and Nelson in \citeJN19, for both PGL₂(ℝ) and PGL₂(ℂ), which is applied to a test vector conjecture for local triple product periods.

Citations

• Uniform subconvexity bounds for GL(2)× GL(2) L-functions in the spectral aspect
• Subconvexity for Rankin Selberg L-Functions at Special Points
• Local integral transforms and global spectral decomposition
• The subconvexity problem for Rankin-Selberg and triple product L-functions
• The Weyl bound for triple product L-functions
• Periods and Reciprocity II
• Eisenstein series and the cubic moment for PGL(2)
• Hida families and p-adic triple product L-functions
• Triple product formula and mass equidistribution on modular curves of level N
• The circle method and bounds for L-functions - III: t-aspect subconvexity for GL(3) L-functions
• Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels
• On the Ramanujan conjecture over number fields
• Rankin Triple Products and Quantum Chaos
• Weak subconvexity for central values of L-functions
• Mass equidistribution for Hecke eigenforms
• Subconvexity bounds for triple L-functions and representation theory
• Sparse equidistribution problems, period bounds, and subconvexity
• Harmonic Analysis

Related

• The subconvexity problem for Rankin-Selberg and triple product L-functions
• A new subconvex bound for \rm GL(3) L-functions in the t-aspect
• Spectral Reciprocity and Hybrid Subconvexity Bound for triple product L-functions
• Subconvexity for the Rankin-Selberg L-function in both levels
• Subconvexity for Rankin Selberg L-Functions at Special Points