The Second Moment of ${\mathrm{GL}}_3\times {\mathrm{GL}}_2$ $L$-functions at Special Points

Abstract

Let $\varphi$ be a fixed Hecke--Maass form for ${\mathrm{SL}}_3(\mathbb{Z})$ and $u_j$ traverse an orthonormal basis of Hecke--Maass forms for ${\mathrm{SL}}_2(\mathbb{Z})$. Let $1/4+t_j^2$ be the Laplace eigenvalue of $u_j$. In this paper, we prove the mean Lindelöf hypothesis for the second moment of $L(1/2+i{t}_j,\varphi \times u_j)$ on $T<t_j⩽T+\sqrt{T}$. Previously, this was proven by Young on $t_j⩽T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein--Kloosterman' cancellation.

22 pages