The Weyl bound for Rankin-Selberg $L$-functions with Joint Ramification

TL;DR

This paper proves the Weyl bound for Rankin-Selberg L-functions in a joint ramification setting using advanced trace formulas and p-adic analysis, advancing understanding of automorphic L-functions.

Contribution

It introduces a refined Petersson trace formula and a specialized Voronoi summation formula to establish new bounds for Rankin-Selberg L-functions.

Findings

Established the Weyl bound in a joint ramification context

Developed a new p-adic stationary phase method for Whittaker functions

Provided sharp bounds for integrals of Whittaker function products

Abstract

In this paper, we establish the Weyl bound for the Rankin-Selberg $L$-function in a certain joint ramification setting. To achieve this result, we employ the refined Petersson trace formula and develop a special Vorono\"i summation formula. Additionally, we obtain the sharp bound for the integral of products of Whittaker functions via the $p$-adic stationary phase method.