Automorphic Representations and L-Functions for the General Linear Group

Abstract

This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.

On periodicity of geodesic continued fractions

“…For a positive integer n and a prime number p, let h n = h n ∞ := GL n (R)/R × O(n) and let h n p := GL n (Q p )/Q × p GL n (Z p ). See Goldfeld [9], [10], Terras [17], [18], or Borel [2] for basic references. For S, a finite set of places of Q, let h n S be the generalized S-upper half space v∈S h n v .…”

The role of the Ramanujan conjecture in analytic number theory

“…The Ramanujan conjecture for GL n We now pass to the general setting of the group GL n , defined over a number field F . We refer to [54] for an introduction to this theory and some relevant notation. A dictionary between classical language and adelic language can be found in [49].…”

Modular Forms, a Computational Approach

“…Very explicit formulas for Eisenstein series on GL 3 can be seen in [Bum84]. For a down-to-earth exposition of some of the Eisenstein series on GL n , we refer to [Gol05].…”