By Dorian Goldfeld, Joseph Hundley
This graduate-level textbook presents an user-friendly exposition of the idea of automorphic representations and L-functions for the overall linear staff in an adelic environment. The authors hold definitions to a minimal and repeat them whilst reintroduced in order that the ebook is obtainable from any access element, and with out earlier wisdom of illustration idea. additionally they comprise concrete examples of either worldwide and native representations of GL(n), and current their linked L-functions. the idea is built from first ideas for GL(1), then conscientiously prolonged to GL(2) with entire specific proofs of key theorems. a number of of the proofs are right here awarded for the 1st time, together with Jacquet's basic and chic facts of the tensor product theorem. eventually, the better rank state of affairs of GL(n) is given an in depth remedy. Containing a number of routines, this ebook will inspire scholars and researchers to start operating during this fertile box of analysis.
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Additional info for Automorphic Representations and L-Functions for the General Linear Group: Volume 1
Example text
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 . Define the rational function field adeles to be the restricted product (relative to the subgroups O Q ) of AF p (T ) = F p ((1/T )) × Fq ((Q)). Q∈F p [T ] monic irreducible Show that a fundamental domain for the additive action of F p (T ) on AF p (T ) is given by 1 O∞ × T OQ . 13* In this exercise we indicate all of the necessary adjustments in order to make sense of Fourier transforms, the Fourier inversion theorem, and the Poisson summation formula for the function field adeles AF p (T ) .
Of Q p which are pairwise disjoint, ∞ such that Un is still compact. n=1 It is easy to see that a measure μ on Q p must satisfy p−1 μ a + pn Z p = μ a + bp n + p n+1 Z p b=0 for all 0 ≤ a ≤ p − 1 and n ∈ Z, because the compact set on the left side is the disjoint union of the ones on the right. 2 (Locally constant function) A function f : Q p → C is said to be locally constant on a subset V ⊂ Q p if for every x ∈ V there exists an open set U ⊂ V containing x such that f (x) = f (u) for all u ∈ U.
10. 13. Recall that the symbol v is allowed to denote either ∞ or a non-trivial monic irreducible polynomial Q ∈ F p [T ]. The degree of v, denoted deg(v), is given by deg(v) = if v = ∞, 1, deg(Q), if v = Q is a monic irreducible polynomial. v D= (n v ∈ Z). v The degree of the divisor D is the integer deg(D) = We let L(D) = v n v deg(v). f ∈ F p (T ) | f |v ≤ p n v deg(v) for all v . A rational function f ∈ L(D) is bounded v-adically in terms of the divisor D. We will see that L(D) is an F p -vector space, and the essence of the Riemann-Roch theorem is that we can calculate its dimension.
