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

TL;DR

This paper simplifies the proof of subconvexity bounds for certain $L$-functions associated with holomorphic cusp forms and Dirichlet characters, using a modified delta method and second moment techniques.

Contribution

It introduces a simplified proof of subconvexity bounds for $GL(2) imes GL(1)$ $L$-functions employing a modified delta method and unamplified second moment approach.

Findings

Established subconvexity bounds for $L(1/2,f\otimes\chi)$

Utilized a modified delta method for the proof

Achieved bounds under specific level and conductor conditions

Abstract

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^\eta$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes\chi)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $\chi$ 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.