Hybrid Level Aspect Subconvexity for $GL(2)\times GL(1)$ Rankin-Selberg $L$-Functions

Abstract

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^{\hat{I}}\cdot$ with $0<\hat{I}\cdot <2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes \ddot{I}‡)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $\ddot{I}‡$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.

Correct bounds for j-Bessel functions and add definition of Kloosterman sum associated to cups. Main result is not changed