Eisenstein series, uniqueness principle, and finite order bounds

The main purpose of this thesis is to give a new proof of the finiteness of order of the constant term coefficients of Eisenstein Series on higher rank groups. The method is applicable to any semisimple group with a few minor assumptions. The bounds are derived from the analytic continuation of Eisenstein Series. The analytic continuation follows an idea originally introduced by Selberg in the case of rank one groups and developed by others in higher rank. The first part of the proof reduces the analytic continuation of the Eisenstein Series to the analytic continuation of their constant terms coefficients. The main tool that is used is the Fredholm determinant, which allows one to control the size of the auxiliary functions that arise. The coefficients are then analytically continued by using the eigenfunction properties of the Eisenstein Series itself. Using the fact that the Eisenstein Series is an eigenfunction for the ring of bi-invariant differential operators leads to an infinite system of linear equations with the constant term coefficients as unknowns. An effective version of the Uniqueness Principle is used to extract a finite number of linear equations from the infinite system so that the corresponding determinant is bounded away from zero by a controlled amount. In order to prove finite order bounds, uniform (in the group variable) estimates on the coefficients in the infinite linear system are established. This uniformity allows one to extract bounds from the Uniqueness Principle even though there is no information on which finite set of linear equations is being used.