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

Abstract

Let $\pi$ be a Hecke-Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ and $f$ be a holomorphic cusp form for $\mathrm{SL}(2,\mathbb{Z})$ of weight $k$ or a Hecke-Maass cusp form corresponding to the Laplacian eigenvalue $1/4+k^2$, $k\ge 1$, for $\mathrm{SL}(2,\mathbb{Z})$. In this paper, we prove the following subconvexity bound \begin{align*} L\left({1}/{2},\ π\times f\right) \ll_{π,\,ε} k^{\frac{3}{2}-\frac{1}{51}+ε}. \end{align*}

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