On the subconvexity problem for $GL(3)\times GL(2)$ $L$-functions

TL;DR

This paper establishes a subconvexity bound for the Rankin-Selberg $L$-function $L(s, g\times f)$ at the central point in the level aspect, assuming a lower bound on a short sum of Fourier coefficients of $f$.

Contribution

It proves a subconvexity bound for $L(s, g\times f)$ in the $q$ aspect under a new conditional assumption on Fourier coefficients.

Findings

Conditional subconvexity bound established for $L(s, g\times f)$

Relates Fourier coefficient bounds to $L$-function behavior

Advances understanding of $GL(3)\times GL(2)$ $L$-functions

Abstract

Fix $g$ a self-dual Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $f$ be a holomorphic newform of prime level $q$ and fixed weight. Conditional on a lower bound for a short sum of squares of Fourier coefficients of $f$, we prove a subconvexity bound in the $q$ aspect for $L(s, g\times f)$ at the central point.