Subconvexity Bounds for Rankin-Selberg L-Functions via the Spectral Theory of Automorphic Forms

This monograph establishes a rigorous subconvexity bound for the Rankin-Selberg L-function L(s, ') on the critical line Re(s) = 1/2, where and ' are cuspidal automorphic representations of GL(2, AQ). By invoking the spectral theory of automorphic forms, specifically the Kuznetsov trace formula and the amplification method, we surpass the convexity barrier imposed by the Phragmén-Lindelöf principle. The analysis necessitates a precise estimation of shifted convolution sums involving Fourier coefficients of Maass forms. We employ Voronoi summation formulas on GL(2) and stationary phase analysis of the resulting Bessel transforms to control the off-diagonal geometric terms. The main result yields a bound of the form C(')14 - for an explicit > 0, where C(') denotes the analytic conductor.