The Second Moment of Rankin-Selberg L-function and Hybrid Subconvexity Bound

Abstract

Let $M,N$ be coprime square-free integers. Let $f$ be a holomorphic cusp form of level $N$ and $g$ be either a holomorphic or a Maaß form with level $M$. Using a large sieve inequality, we establish a bound of the form ${\sum }_g{\left|L^{(j)}\left(1/2+it,f\otimes g\right)\right|}^2{\ll }_{t}M+M^{2/3-\beta }{N}^{4/3}$ where $\beta \approx 1/500$. As a consequence, we obtain subconvexity bounds for $L^{(j)}\left(1/2+it,f\otimes g\right)\ll (MN{)}^{1/2-\alpha }$ for any $N<M$ satisfying the conditions above without using amplification methods. Moreover, by the symmetry, we establish a level aspect hybrid subconvexity bound for the full range when both forms are holomorphic.

References in corpus (1)

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