A new bound k2/3+ε for Rankin-Selberg L-functions for Hecke congruence subgroups

Abstract

Let f be a holomorphic Hecke eigenform for Γ0(N) of weight k, or a Maass eigenform for Γ0(N) with Laplace eigenvalue 1/4 + k2. Let g be a fixed holomorphic or Maass cusp form for Γ0(N). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f ⊗ g) is proved in the k-aspect: L(1/2 + it, f ⊗ g) ≪N,g,t,ε k2/3+ε, while a convexity bound is only ≪ k1+ε. The dependence of the implied constant on t and the level N is polynomial. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to Res > -1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.