By Dorian Goldfeld, Joseph Hundley

This graduate-level textbook presents an simple exposition of the speculation of automorphic representations and L-functions for the final linear staff in an adelic surroundings. The authors hold definitions to a minimal and repeat them whilst reintroduced in order that the publication is out there from any access element, and with out previous wisdom of illustration conception. additionally they contain concrete examples of either international and native representations of GL(n), and current their linked L-functions. the idea is built from first rules for GL(1), then conscientiously prolonged to GL(2) with entire distinct proofs of key theorems. numerous of the proofs are the following awarded for the 1st time, together with Jacquet's basic and chic evidence of the tensor product theorem. eventually, the better rank state of affairs of GL(n) is given an in depth therapy. Containing a number of workouts, this e-book will encourage scholars and researchers to start operating during this fertile box of analysis.

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**Extra resources for Automorphic Representations and L-Functions for the General Linear Group: Volume 1**

**Sample text**

1 of a fundamental domain. Next, we consider the multiplicative action of Q× on the ideles A× Q which we denote by · which is defined by q · x = {q · x∞ , q · x2 , q · x3 , . . } × for all x = {x∞ , x2 , x3 , . . } ∈ A× Q and q ∈ Q . Here q · x v denotes multiplication in Qv . 6 (Strong approximation for ideles) A fundamental domain D for Q× \A× Q is given by D = {x∞ , x2 , x3 , . . } 0 < x∞ < ∞, x p ∈ Z×p for all finite primes p Z×p . = (0, ∞) · p That is, we have A× Q = α · D, (disjoint union).

Prove that the completion of F p (T ) with respect to the absolute value | · |∞ can be identified with F p ((1/T )). (d) Let the symbol v denote either ∞ or a monic irreducible polynomial Q ∈ F p [T ]. Show that the following product formula holds for any non-zero rational function f ∈ F p (T ): | f |v = 1. v Exercises for Chapter 1 35 (e) For each monic irreducible polynomial Q ∈ F p [T ], we define a subring of the completion by OQ = f ∈ Fq ((Q)) | f | Q ≤ 1 . Similarly, we have O∞ = f ∈ F p ((1/T )) | f |∞ ≤ 1 .

C < C (1 + |t|∞ ) M for all t ∈ R. 2. 3. 4. If c1 , c2 ∈ C, are arbitrary complex constants, and φ1 , φ2 ∈ Sω , then it is easy to see that c1 φ1 + c2 φ2 is again automorphic with character ω. The space Sω is, therefore, a vector space over C. Setting g = {1, 1, 1, . . 4 (2) that φ(z) = cω(z), with c = φ({1, 1, 1, . . }). Thus Sω is a one-dimensional space. 2? The reason is that we want to give a uniform definition of automorphic form for G L(n, AQ ) that holds for all n = 1, 2, 3, . .. 1 Automorphic forms for GL(1, AQ ) 41 both lie in the same space.