Subconvexity Bounds for Rankin-Selberg L-Functions and Quantum Unique Ergodicity on Arithmetic Surfaces

This monograph investigates the intrinsic connection between the subconvexity problem for Rankin-Selberg L-functions and the Quantum Unique Ergodicity (QUE) conjecture for arithmetic surfaces. We analyze the asymptotic distribution of mass for high-energy Hecke-Maass eigenfunctions on the modular surface X = SL(2, Z) H. Central to our exposition is Watson's formula, which relates the triple product period of eigenfunctions to the central value of the associated triple product L-function. We demonstrate that establishing a subconvex bound for the Rankin-Selberg L-function L(1/2, g) in the spectral aspect is a sufficient condition for the equidistribution of the probability measures d = |(z)|2 dvol(z) as the Laplacian eigenvalue . We review recent breakthroughs by Lindenstrauss, Soundararajan, and Holowinsky, and discuss the implications of the Generalized Lindelöf Hypothesis in this context.