Uniform Boundedness of Pole Order of General Eisenstein Series

Abstract

This work consists of two parts. In the first part we give a general introduction to the chapter 7 of [L1], and settle down some bases needed for the second chapter in which we prove that the order of the poles of a residual Eisenstein series on an arbitrary reductive group $G$ which satisfies the conditions of the chapter 1 of this work is uniformly bounded by a constant which depends only on the number of elements of a subgroup of the Weyl group of $G$ via the methods developed in [F1] and [F2]. Having a general understanding of the main assertions and difficulties that Langlands had faced and solved through his treatment of Eisenstein series is crucial in understanding [F1] and [F2], on them this work has been built, consequently we start this work with an introduction to Eisenstein series and afterwards in chapter 1 we review Eisenstein systems, and in chapter two we will prove the main claim of this work.