Subconvexity bounds for Rankin–Selberg L-functions for congruence subgroups

Estimation of shifted sums of Fourier coefficients of cusp forms plays crucial roles in analytic number theory. Its known region of holomorphy and bounds, however, depend on bounds toward the general Ramanujan conjecture. In this article, we extended such a shifted sum meromorphically to a larger half plane Res>1/2Res>1/2 and proved a better bound. As an application, we then proved a subconvexity bound for Rankin–Selberg L-functions which does not rely on bounds toward the Ramanujan conjecture: Let f be either a holomorphic cusp form of weight k  , or a Maass cusp form with Laplace eigenvalue 1/4+k21/4+k2, for Γ0(N)Γ0(N). Let g be a fixed holomorphic or Maass cusp form. What we obtained is the following bound for the L  -function L(s,f⊗g)L(s,f⊗g) in the k aspect:L(1/2+it,f⊗g)≪k1−1/(8+4θ)+ε,L(1/2+it,f⊗g)≪k1−1/(8+4θ)+ε, where θ   is from bounds toward the generalized Ramanujan conjecture. Note that a trivial θ=1/2θ=1/2 still yields a subconvexity bound.