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

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.