New bounds for automorphic L-functions

This dissertation contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation \pi of GL(m) over a number field. The approximation involves a smooth truncation of the Dirichlet series L(s,\pi) and L(s,\tilde{\pi}) after about \sqrt{C} terms, where C denotes the analytic conductor (of \pi and \tilde{\pi} at the central point) introduced by Iwaniec and Sarnak. We investigate the decay rate of the cutoff function and its derivatives. We also see that the truncation can be made uniformly explicit at the cost of an error term. The results extend to products of central values. We establish, via the Hardy–Littlewood circle method, a nontrivial bound on shifted convolution sums of Fourier coefficients coming from classical holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. We achieve polynomial uniformity in all the parameters of the cusp forms by carefully estimating the Bessel functions that enter the analysis. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms. We also study the shifted convolution sums via the Sarnak–Selberg spectral method. For holomorphic cusp forms this approach detects optimal cancellation over any totally real number field. For Maass cusp forms the method is burdened with complicated integral transforms. We succeed in inverting the simplest of these transforms whose kernel is built up of Gauss hypergeometric functions