Spectral aspect subconvex bounds for ${\rm U}_{n+1} \times {\rm U}_{n}$

TL;DR

This paper establishes new subconvex bounds for certain automorphic L-functions associated with unitary groups, advancing understanding of their growth and distribution in the context of the Gan--Gross--Prasad conjecture.

Contribution

It proves a subconvexity bound for L-functions of automorphic representations on unitary groups, using refined microlocal analysis and trace formula techniques.

Findings

Subconvex bound for L((    ),1/2) with explicit 

Refinement of microlocal calculus for Lie group representations

Application of geometric side analysis of the relative trace formula

Abstract

Let $(\pi,\sigma)$ 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 \[ L(\pi \times \sigma,1/2) \ll C(\pi \times \sigma)^{1/4 - \delta} \] holds for any fixed \[ \delta < \frac{1}{8 n^5 + 28 n^4 + 42 n^3 + 36 n^2 + 14 n}. \] 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.