Spectral aspect subconvex bounds for \rm U_n+1 × \rm U_n

2020/12/03 by Paul D. Nelson, Nelson, Paul D.
Mathematics · #11F67 #11F70 #11M99 #35A27 #53D50 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2012.02187

openalex publication_date 2020/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (π,σ) traverse a sequence of pairs of cuspidal automorphic representations of a unitary Gan--Gross--Prasad pair (\rm U_n+1,\rm U_n) over a number field, with \rm U_n anisotropic. We assume that at some distinguished archimedean place, the pair stays away from the conductor dropping locus, while at every other place, the pair has bounded ramification and satisfies certain local conditions (in particular, temperedness). We prove that the subconvex bound L(π× σ,1/2) ≪ C(π× σ)^{1/4 - δ} holds for any fixed δlt; (1)/(8 n⁵ + 28 n⁴ + 42 n³ + 36 n² + 14 n). Among other ingredients, the proof employs a refinement of the microlocal calculus for Lie group representations developed with A. Venkatesh and an observation of S. Marshall concerning the geometric side of the relative trace formula.

Citations

• A new subconvex bound for \rm GL(3) L-functions in the t-aspect
• Algebraic twists of GL₃× GL₂ L-functions
• Subconvex bound for \textrmGL(3) × \textrmGL(2) L-functions: \textrmGL(3)-spectral aspect

Related

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• Subconvex bounds for U_n+1× U_n in horizontal aspects
• Relative Trace Formula and Twisted L-functions: the Burgess Bound
• On the spectral aspect density hypothesis and application
• Shifted convolution sums and Burgess type subconvexity over number fields