Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $GL(3)$ spectral aspect

TL;DR

This paper establishes new subconvexity bounds for $GL(3) imes GL(2)$ and $GL(3)$ $L$-functions in the spectral aspect by analyzing amplified second moments of these functions.

Contribution

It introduces a novel amplified second moment method to achieve subconvexity bounds for $GL(3) imes GL(2)$ and $GL(3)$ $L$-functions in the spectral aspect.

Findings

Proves $L(1/2, ext{ extit{pi}} imes f) ext{ bound} \\ll ext{spectral parameter}^{3/2 - 1/2022 + ext{epsilon}}$.

Derives a subconvexity bound for $L(1/2, ext{ extit{pi}})$ as $ ext{spectral parameter}^{3/4 - 1/4044 + ext{epsilon}}$.

Employs amplified second moments to break the convexity barrier in the spectral aspect.

Abstract

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $\mu$ in generic position i.e. away from Weyl chamber walls and away from self dual forms. We study an amplified second moment $\sum_{j} A(\pi_j)|L(1/2,\pi_j\times f)|^2$ and deduce the subconvexity bound \begin{equation*} L(1/2,\pi\times f)\ll_{f,\epsilon} \|\mu\|^{3/2-1/2022+\epsilon}. \end{equation*} As a corollary, when $f=E(z,1/2)$, we also obtain the subconvexity bound \begin{equation*} L(1/2,\pi)\ll_{\epsilon} \|\mu\|^{3/4-1/4044+\epsilon}. \end{equation*}