Hybrid bounds for Rankin-Selberg L-functions

Let M be a square-free integer and P be a prime such that (P,M)=1. We prove a new hybrid bound for L(12,f⊗g) where f is a primitive holomorphic cusp form of level M and g a primitive (either holomorphic or Maass) cusp form of level P satisfying P∼Mη with 0<η<2/15. Particularly in the range β<η<(2−32β)/15 with β=11/4875 we present a strengthened level aspect hybrid subconvexity bound for L(12,f⊗g) relative to the current bounds obtained by Holowinsky-Munshi [11] and Ye [27].