First moment of Rankin-Selberg central L-values and subconvexity in the level aspect

TL;DR

This paper estimates the average size of central Rankin-Selberg L-values for forms of varying levels, leading to subconvex bounds in the level aspect, especially for dihedral forms.

Contribution

It provides a new first moment estimate for Rankin-Selberg L-values in the level aspect, deriving subconvex bounds and extending to derivatives under non-negativity assumptions.

Findings

Bound L(1/2,f×g) by (N + √M) N^ε M^ε for dihedral g

First moment method yields subconvexity in level aspect

Extension to derivatives under non-negativity assumption

Abstract

Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and trivial nebentypus and $g$ is a given form of level $M$. As a result, we recover the bound $ L(1/2,f\times g) \ll_\varepsilon (N + \sqrt{M}) N^\varepsilon M^\varepsilon $ when $g$ is dihedral. The first moment method also applies to the special derivative $L'(1/2,f\times g)$ under the assumption that it is non-negative for all $f\in S^*_k(N)$.