By Dorian Goldfeld, Joseph Hundley

ISBN-10: 1107007992

ISBN-13: 9781107007994

This graduate-level textbook presents an undemanding exposition of the idea of automorphic representations and L-functions for the overall linear team in an adelic atmosphere. Definitions are saved to a minimal and repeated while reintroduced in order that the e-book is out there from any access element, and without previous wisdom of illustration thought. The booklet contains concrete examples of worldwide and native representations of GL(n), and offers their linked L-functions. In quantity 1, the speculation is built from first ideas for GL(1), then rigorously prolonged to GL(2) with entire precise proofs of key theorems. a number of proofs are provided for the 1st time, together with Jacquet's uncomplicated and chic evidence of the tensor product theorem. In quantity 2, the better rank state of affairs of GL(n) is given a close remedy. Containing a number of workouts, this publication will encourage scholars and researchers to start operating during this fertile box of study

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**Additional resources for Automorphic Representations and L-Functions for the General Linear Group: Volume 2**

**Example text**

It follows easily from the properties of e that ψα is continuous and trivial on Un (Q). This rest is given in the next lemma. 2 satisfies ψα (u · u ) = ψα (u) · ψα (u ) for all u, u ∈ Un (AQ ). Proof The lemma follows immediately from the matrix identity ⎛1 ⎜ ⎜ ⎜ ⎜ ⎝ u 1,2 1 u 2,3 .. ∗ .. 1 u n−1,n 1 ⎞ ⎛1 ⎟ ⎜ ⎟ ⎜ ⎟·⎜ ⎟ ⎜ ⎠ ⎜ ⎝ ⎞ u 1,2 1 u 2,3 .. ∗ .. 1 u n−1,n 1 ⎛ 1 u 1,2 + u 1,2 ⎜ 1 ⎜ ⎜ =⎜ ⎜ ⎝ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ u 2,3 + u 2,3 .. ∗ .. 1 u n−1,n + u n,n−1 1 ⎟ ⎟ ⎟ ⎟, ⎟ ⎠ and the fact that e : Q\AQ → C is an additive character.

0 ϕ g Q\AQ = ϕβ·γ (g). 9) that ϕ0 = 0, where 0 = (0, . . , 0) is the zero vector. 5 The Fourier expansion of adelic automorphic forms 35 γ ∈ G L(n − 1, Q) such that β = (0, . . , 0, αn−1 ) · γ . Furthermore, γ is unique up to an element of the stabilizer of (0, . . , 0, αn−1 ), which is Pn−1 (Q). ,0,αn−1 ) (γ g). ,0,αn−1 ) h 0 0 1 ∀h ∈ G L(n − 1, AQ ) . 2, we may define a character of ψ : Un (Q)\Un (AQ ) → C by ψ u 0 y 1 = ψ (u )e p (−αn+1 u), (∀u ∈ Un−1 (AQ ), y ∈ Mat(n − 1 × 1, AQ )). It is then immediate from the definitions that ϕg,αn−1 (u h)(ψ )−1 (u ) du Un−1 (Q)\Un−1 (AQ ) ϕ u = h 0 0 1 g ψ −1 (u) du.

For an integer n ≥ 1, let G L(n, Afinite ) denote the multiplicative subgroup of all afinite ∈ G L(n, AQ ) of the form afinite = {In , g2 , g3 , g5 , ... , } where g p ∈ G L(n, Q p ) for all finite primes p and g p ∈ G L(n, Z p ) for all but finitely many primes p. Here In denotes the n × n identity matrix. We shall define an automorphic representation for G L(n, AQ ) as the vector space Aω (G L(n, AQ )) with three natural actions defined as follows. • Action of the finite adeles G L(n, Afinite ) by right translation: We define an action πfinite : G L(n, Afinite ) → G L Aω (G L(n, AQ )) as follows.

### Automorphic Representations and L-Functions for the General Linear Group: Volume 2 by Dorian Goldfeld, Joseph Hundley

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