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

Abstract

For $M_1$ and $ M_2$ two distinct primes, let $ H_k^\star(M_1M_2, ψ)$ denote the set of primitive newforms of level $M_1M_2$, weight $k\geq 3$ and Nebentypus $ψ$ of conductor $M_1$. Let $π$ 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 $M_2\leq M_1^{1+ε}$.

Theorems & Definitions (28)

• Theorem 1
• Lemma 1: Duality principle
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• Lemma 6