Subconvex bound for $\textrm{GL(3)} \times \textrm{GL(2)}$ $L$-functions: $\textrm{GL(3)}$-spectral aspect

Abstract

Let $\phi$ be a Hecke-Maass cusp form for $\mathrm{SL(3, \mathbb{Z})}$ with Langlands parameters $({\bf t}_{i})_{i=1}^{3}$ and $f$ be a holomorphic or Hecke-Maass cusp form for $\mathrm{SL(2,\mathbb{Z})}$. In this article, we prove the following subconvex bound $$ L\left(\phi \times f, 1/2\right) \ll_{f,\epsilon} T^{ \frac{3}{2}-\delta_\xi+\epsilon},\ \delta_\xi=\min\{\xi/4, \, (1-2\xi)/4 \}, $$ for the central value $ L\left(\phi \times f, 1/2\right) $ in the $\mathrm{GL(3)}$-spectral aspect, where $({\bf t}_{i})_{i=1}^{3}$ satisfies $$|{\bf t}_{3} - {\bf t}_{2}| \asymp T^{1-\xi }, \quad \, {\bf t}_{i} \asymp T, \quad \, \, i=1,\,2,\,3,$$ with $\xi$ a real number such that $0 < \xi <1/2$.