Number Theory

Automorphic Representations and L-Functions for the General by Dorian Goldfeld, Joseph Hundley

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By Dorian Goldfeld, Joseph Hundley

This graduate-level textbook offers an effortless exposition of the speculation of automorphic representations and L-functions for the final linear crew in an adelic atmosphere. The authors preserve definitions to a minimal and repeat them while reintroduced in order that the ebook is on the market from any access element, and without previous wisdom of illustration idea. in addition they comprise concrete examples of either international and native representations of GL(n), and current their linked L-functions. the speculation is built from first ideas for GL(1), then rigorously prolonged to GL(2) with whole specified proofs of key theorems. a number of of the proofs are the following offered for the 1st time, together with Jacquet's uncomplicated and chic evidence of the tensor product theorem. eventually, the better rank scenario of GL(n) is given an in depth therapy. Containing quite a few workouts, this ebook will encourage scholars and researchers to start operating during this fertile box of study.

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Example text

Step 2: We show there exists fixed N ∈ Z such that f α = 0 unless α = Nn with n ∈ Z. We may think of the minimal positive N satisfying this condition as the conductor of f . 8 Fourier expansion of periodic adelic functions 29 unless |α| p ≤ 1. Since h p = 1Z p is the characteristic function of Z p for all but finitely many primes p, it follows that h(α) = 0 unless α = Nn (with n ∈ Z) where N = i=1 piai and p1 , p2 , . . , p are the finitely many primes where h pi =/ 1Z pi for i = 1, 2, . . , .

To get h 0 , we first need a Schwartz function h 0,∞ : R → R satisfying ∞ h 0,∞ (x + n) = 1, (∀ x ∈ R). 7) n=−∞ Such a function may be constructed, for example, by letting g(x) be any smooth function such that supp (g) = [− 14 , ∞) and g(x) = 1 for all x ≥ 0. Then put h 0,∞ (x) = g(x) − g(x − 1). Now, for x = {x∞ , . . , x p , . . } ∈ AQ define h 0 (x) = h 0,∞ (x∞ )· p 1Z p (x p ) where 1Z p denotes the characteristic function of Z p . 6), for any x ∈ R × p Z p . Any other x may be written as β + y for y ∈ R × p Z p and β ∈ Q, so one has only to make a change of variables in the summation.

Since F p [T ] is a unique factorization domain, any non-zero rational function f ∈ F p (T ) can be written uniquely as f = Q r g for some integer r and some rational function g such that Q does not divide the numerator or denominator of g. Define an absolute value on F p (T ) by | f | Q = p −r deg(Q) and |0| Q = 0. (ii) Let f = P1 /P2 be a non-zero rational function. Define | f |∞ = p deg(P1 )−deg(P2 ) , where deg denotes the degree of a polynomial. Set |0|∞ = 0. (b) Let Q be a monic irreducible polynomial with coefficients in F p , let d = deg(Q), and let q = p d .

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