Stationary phase analysis for analytic newvectors and application to subconvexity problems

TL;DR

This paper develops a stationary phase analysis for analytic newvectors to improve bounds on triple product and Rankin-Selberg L-functions, achieving subconvexity results in the spectral aspect with conductor dropping.

Contribution

It extends previous work by applying stationary phase analysis to analytic newvectors for PGL2(R) and PGL2(C), leading to new subconvexity bounds with conductor dropping.

Findings

Established upper bounds for triple product and Rankin-Selberg L-functions.

Achieved subconvexity bounds in the spectral aspect.

Introduced a stationary phase analysis for analytic newvectors.

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 $$L(\pi_1 \otimes \pi_2 \otimes \pi_3,\frac{1}{2})\ll_{\pi_3,\epsilon}C(\pi_1\otimes\pi_2)^{\frac{1}{2} + \epsilon} \left( \frac{C(\pi_1 \otimes \pi_2)}{C(\pi_2 \otimes \pi_2)}\right)^{-\delta}$$ in the spectral aspect, allowing conductor dropping. In particular, we obtain a subconvexity bound when $\pi_1\otimes\pi_2$ 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 \cite{JN19}, for both $\mathrm{PGL}_2(\mathbb{R})$ and $\mathrm{PGL}_2(\mathbb{C})$, which is applied to a test vector conjecture for local triple product periods.