By Dorian Goldfeld, Joseph Hundley

This graduate-level textbook offers an easy exposition of the speculation of automorphic representations and L-functions for the overall linear workforce in an adelic environment. The authors hold definitions to a minimal and repeat them whilst reintroduced in order that the e-book is offered from any access element, and without past wisdom of illustration thought. in addition they contain concrete examples of either worldwide and native representations of GL(n), and current their linked L-functions. the speculation is constructed from first rules for GL(1), then rigorously prolonged to GL(2) with entire exact proofs of key theorems. numerous of the proofs are the following awarded for the 1st time, together with Jacquet's uncomplicated and chic evidence of the tensor product theorem. ultimately, the better rank state of affairs of GL(n) is given an in depth remedy. Containing a variety of routines, this ebook will encourage scholars and researchers to start operating during this fertile box of analysis.

**Read Online or Download Automorphic Representations and L-Functions for the General Linear Group: Volume 1 PDF**

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**Example text**

Let S = {∞, p1 , p2 , . . , p } denote the finite set of primes such that x p ∈ Z p and h p is the characteristic function of Z p for p ∈ S. If α ∈ Q, it follows that h p (x p + α) = 0 for p ∈ S unless α is an integer in Z p . Since h v is a locally Adeles over Q 24 constant compactly supported function for v ∈ S this implies that there exists a rational integer M so that h(x + α) = 0 unless α = Mn with n ∈ Z. Therefore (at least formally) n . h(x + α) = h x+ M α∈Q n∈Z Finally, for fixed x ∈ AQ we must have that h x + Mn has rapid decay in n as n → ±∞.

Remarks A fundamental domain is precisely a choice of one point from each orbit of G. If G\X is the quotient space with the quotient topology and π : X → G\X is the quotient map, then the fundamental domain is the image of a section σ : G\X → X . ) The construction of an explicit fundamental domain for the action of the additive group Q on the adele group AQ is equivalent to a generalization of the ancient Chinese remainder theorem. 2 (Chinese Remainder Theorem) Let p1 , p2 , . . pn be distinct primes.

Hint: It may be useful to know that an infinite product (1 + an ) converges if the series an converges absolutely. 9 Show that a smooth adelic function f : AQ → C is continuous (for the adelic topology). 1. Prove that e(αu) du = Q\AQ 1, if α = 0, 0, if α = 0. 10. 10). We will show there is a smooth periodic adelic function g : AQ → C such that g(x) = f (x) − f (0) , 1 − e(x/N ) (†) whenever this expression makes sense. 10 to assume that each finite factor of h is of the form h p = 1a+ pm Z p for some a ∈ Q p and some integer m (that depend on the prime p).