Second moments of Rankin-Selberg convolution and shifted Dirichlet series

TL;DR

This paper analyzes the second moments of Rankin-Selberg convolutions over $ ext{Gamma}_0(N)$, deriving sharp error bounds and hybrid subconvexity results, and demonstrates the existence of Maass forms with non-zero central L-values.

Contribution

It provides a new spectral moment formula for Rankin-Selberg convolutions with sharp error terms and establishes hybrid subconvexity bounds and simultaneous non-vanishing results.

Findings

Derived a main term plus sharp error for spectral moments of convolutions.

Obtained hybrid Weyl-type subconvexity bounds in $t$ and spectral aspects.

Proved existence of Maass forms with non-zero central L-values for fixed modular forms.

Abstract

In this paper we work over $\Gamma_0(N)$, for any $N$ and write the spectral moment of a product of two distinct Rankin-Selberg convolutions at a general point on the critical line $\frac{1}{2}+it$ as a main term plus a sharp error term in the $t$ aspect and the spectral aspect. As a result we obtain hybrid Weyl type subconvexity results in the $t$ and spectral aspects. Also, for fixed modular forms $f$, $g$ of even weight $k\geq 4$ we show there exists a Maass cusp form $u_j$ such that $L(1/2, f\times u_j)$, $L(1/2, g\times u_j)$ are simultaneous non-zero.