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

Abstract

Let $\ddot{I}•$ be a Hecke-Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ with Langlands parameters $({\mathbf{t}}_i{)}_{i=1}^3$ 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 $$L\left(\ddot{I}•\times f,1/2\right){\ll }_{f,\hat{I}\mu }{T}^{\frac{3}{2}-\hat{I}{´}_{\hat{I}}¾+\hat{I}\mu },\hat{I}{´}_{\hat{I}}¾=\min \{\hat{I}¾/4,(1-2\hat{I}¾)/4\},$$ for the central value $L\left(\ddot{I}•\times f,1/2\right)$ in the $\mathrm{GL}(3)$-spectral aspect, where $({\mathbf{t}}_i{)}_{i=1}^3$ satisfies $$|{\mathbf{t}}_3-{\mathbf{t}}_2|\asymp T^{1-\hat{I}¾},{\mathbf{t}}_i\asymp T,i=1,2,3,$$ with $\hat{I}¾$ a real number such that $0<\hat{I}¾<1/2$.

40 pages