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

By Dorian Goldfeld, Joseph Hundley

This graduate-level textbook offers an basic exposition of the idea of automorphic representations and L-functions for the final linear staff in an adelic atmosphere. The authors retain definitions to a minimal and repeat them while reintroduced in order that the booklet is on the market from any access aspect, and with out past wisdom of illustration thought. in addition they comprise concrete examples of either international and native representations of GL(n), and current their linked L-functions. the speculation is constructed from first rules for GL(1), then conscientiously prolonged to GL(2) with entire precise proofs of key theorems. a number of of the proofs are right here awarded for the 1st time, together with Jacquet's easy and chic facts of the tensor product theorem. ultimately, the better rank state of affairs of GL(n) is given an in depth remedy. Containing quite a few routines, this e-book will inspire scholars and researchers to start operating during this fertile box of study.

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5 p-adic integration 13 is a disjoint union of compact open sets Ui ⊂ Q p such that f is the constant function ci ∈ C on each Ui for i = 1, 2, . . , n. Then we define f (x) dμ (x) = c1 μ (U1 ) + c2 μ (U2 ) + · · · + cn μ (Un ) . A Remarks (1) We shall refer to dμ (x) as the “differential induced from” the measure μ. This is perhaps best thought of as a common and extremely useful abuse of notation. What has been defined rigorously is a linear functional on the space of locally constant, compactly supported functions f.

Here N is the conductor of f as in Step 2. Define a new function g(x) = f (x) . 1 − e (x/N ) By definition, g is again a smooth periodic adelic function. We compute f n N n x dx N f (x)e − = Q\AQ x N 1−e = Q\AQ g(x)e − = Q\AQ g(x)e − n x dx − N n x dx N g(x)e − Q\AQ n−1 x N dx = g Nn −g n−1 . N It follows that f n N g Nn − g n−1 = 0, = n∈Z N n∈Z since the latter is a telescoping sum where all the terms cancel. 4. 7. 2) Exercises for Chapter 1 which holds for any idele y = {y∞ , y2 , y3 , . . } and where |y|A = the adelic absolute value.

This is because h ∞ is a classical Schwartz function and for the finitely many primes v ∈ S the function h v is absolutely bounded. In all other cases h v is either 1 or 0. The stated uniformity can be obtained because M can be chosen independent of x, for x in a compact set. 4 (Smooth adelic function) An adelic function f : AQ → C is said to be smooth if for any point x0 ∈ AQ , there exists an open set U U U : R → C such that f (x) = f ∞ (x∞ ) (containing x0 ) and a smooth function f ∞ for all adeles x = {x∞ , x2 , .

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