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

TL;DR

This paper establishes an upper bound for the second moment of certain $L$-functions associated with automorphic forms, demonstrating Lindelöf-consistent growth in the level aspect for specific primes.

Contribution

It provides a Lindelöf-consistent upper bound for the second moment of $ extrm{GL(3)} imes extrm{GL(2)}$ $L$-functions in the level aspect, extending understanding of their growth.

Findings

Upper bound $oxed{ ext{M}_1^{1+ ext{epsilon}}}$ for the second moment.

Valid in the range $M_2 ext{ } extless ext{ } M_1^{1+ ext{epsilon}}$.

Supports Lindelöf hypothesis consistency in this setting.

Abstract

For $M_1$ and $ M_2$ two distinct primes, let $ H_k^\star(M_1M_2, \psi)$ denote the set of primitive newforms of level $M_1M_2$, weight $k\geq 3$ and Nebentypus $\psi$ of conductor $M_1$. Let $\pi$ be a fixed $SL(3, \mathbb{Z})$ Hecke cusp form. We prove a Lindel\"of--consistent upper bound for the second moment \[ \mathop{ \sum_{\substack{\psi(M_1) \\ \psi(-1)=(-1)^k }}} \sideset{}{^h}\sum_{f \in H_k^{\star}(M_1M_2,\psi)} |L(1/2, \pi \times f)|^2 \ll_{\pi,\epsilon} M_1^{1+\epsilon}\] in the range $M_2\leq M_1^{1+\epsilon}$.