The second moment of $GL(4) \times GL(2)$ $L$-functions at special points

TL;DR

This paper establishes upper bounds for the second moment of certain $L$-functions involving $GL(4)$ and $GL(2)$ at special points, aligning with the Lindel"{o}f hypothesis, extending previous results beyond $n \,\leq\, 3$.

Contribution

It provides the first known upper bounds for the second moment of $GL(4) \times GL(2)$ $L$-functions at special points, advancing the understanding of higher rank automorphic $L$-functions.

Findings

Upper bounds consistent with Lindel"{o}f hypothesis.

Extension of second moment bounds to $GL(4) \times GL(2)$.

Results align with conjectural growth rates.

Abstract

In this paper, we obtain upper bounds for the second moment of $L(u_j \times \phi, \frac{1}{2} + it_j)$, where $\phi$ is a Hecke Maass form for $SL(4, \mathbb Z)$, and $u_j$ is taken from an orthonormal basis of Hecke-Maass forms on $SL(2, \mathbb{Z})$ with eigenvalue $1/4 + t_j^2$. The bounds are consistent with the Lindel\"{o}f hypothesis. Previously these types of upper bounds are available for only $GL(n) \times GL(2)$, where $n \leq 3.$