Shintani functions and automorphic $L$-functions for ${\rm GL}(n)$

L-Functions and Automorphic Representations*

There are two kinds of L-functions, and they will be described below: motivic L-functions which generalize the Artin L-functions and are defined purely arithmetically, and automorphic L-functions, defined by data which are largely transcendental. Within the automorphic L-functions a special class can be singled out, the class of standard L-functions, which generalize the Hecke L-functions and f...

DERIVATIVES AND L-FUNCTIONS FOR GLn

Automorphic Representations and L-functions

• Decomposition by central characters • Square-integrable cuspforms • Smoothness of cuspforms • Eigen-cuspforms and automorphic representation • Dirichlet series versus zeta and L-functions • L-functions defined via local data • Factoring unitary representations of adele groups • Spherical representations and Satake parameters • Local data, L-groups, higher L-functions References and historical...

Lectures on automorphic L-functions

PREFACE This article follows the format of five lectures that we gave on automorphic Lfunctions. The lectures were intended to be a brief introduction for number theorists to some of the main ideas in the subject. Three of the lectures concerned the general properties of automorphic L-functions, with particular reference to questions of spectral decomposition. We have grouped these together as ...

A Polya Operator for Automorphic L-functions

Let k be a number field and letM be a simple k-algebra with center k of rank n, i.e. dimk(M) = n 2. Fix a k-involution x 7→ xτ on M , i.e. (xy)τ = yτxτ as well as (xτ )τ = x and xτ = x when x ∈ k. We will assume that M and τ are defined over the ring of integers O of k. The standard example will be the algebra of n× n matrices Matn(k) and x τ = xt the transposed matrix. For any ringR over k let...