Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $GL(3)$ 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,π\times f)\ll_{f,ε} \|μ\|^{3/2-1/2022+ε}. \end{equation*} As a corollary, when $f=E(z,1/2)$, we also obtain the subconvexity bound \begin{equation*} L(1/2,π)\ll_ε \|μ\|^{3/4-1/4044+ε}. \end{equation*}

Computations for f Eisenstein added. Arguments in stationary phase analysis corrected. Final bound updated

References in corpus (3)

• On the subconvexity problem for L-functions on GL(3)
• Subconvex bound for $\text{GL(3)}\times \text{GL(2)}$ $L$-functions: $\text{GL(3)}$-spectral aspect
• Parametrization of Kloosterman sets and ${\mathrm{SL}}_3$-Kloosterman sums

Cited by in corpus (1)

• Spectral aspect subconvex bounds for ${\mathrm{U}}_{n+1}\times {\mathrm{U}}_n$