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

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 $f\in S_k^{\ast }(N)$ 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 $L(1/2,f\times g){\ll }_{\epsilon }(N+\sqrt{M})N^{\epsilon }{M}^{\epsilon }$ when $g$ is dihedral. The first moment method also applies to the special derivative $L^{\prime }(1/2,f\times g)$ under the assumption that it is non-negative for all $f\in S_k^{\ast }(N)$.

References in corpus (1)

• Level Aspect Subconvexity For Rankin-Selberg $L$-functions