Second moment of $\text{GL(3)}\times \text{GL(2)}$ $L$--functions

Abstract

For $M_1$ and $M_2$ two distinct primes, let $H_k^{\star }(M_1{M}_2,\ddot{I}ˆ)$ denote the set of primitive newforms of level $M_1{M}_2$, weight $k\ge 3$ and Nebentypus $\ddot{I}ˆ$ of conductor $M_1$. Let $\ddot{I}€$ be a fixed $SL(3,\mathbb{Z})$ Hecke cusp form. We prove a Lindelöf--consistent upper bound for the second moment \[ \mathop{ \sum_{\substack{ψ(M_1) \\ ψ(-1)=(-1)^k }}} \sideset{}{^h}\sum_{f \in H_k^{\star}(M_1M_2,ψ)} |L(1/2, π\times f)|^2 \ll_{π,ε} M_1^{1+ε}\] in the range $M_2\le M_1^{1+\hat{I}\mu }$.

23 pages