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

Sumit Kumar

Published 2020-07-09Version 1

Let $\pi$ be a Hecke-Maass cusp form for $SL(3, \mathbb{Z})$ and $f$ be a holomorphic cusp form with weight $k$ or Hecke-Maass cusp form corresponding to the Laplacian eigenvalue $1/4+k^2$, $k\geq 1$ for $SL(2,\mathbb{Z})$. In this paper, we prove the following subconvexity bound: L\left(\frac{1}{2}, \pi \times f\right) \ll_{\pi,\epsilon} k^{\frac{3}{2}-\frac{1}{51}+\epsilon}, for the central values $L(1/2,\pi \times f)$ of the Rankin-Selberg $L$-function of $\pi$ and $f$. Using the same method, by taking $\pi$ to be the Eisenstein series, we also obtain the following subconvexity bound: L\left( \frac{1}{2},\ f\right) \ll_{\epsilon} k^{\frac{1}{2}-\frac{1}{153}+\epsilon}.