N o t a t i o n a n d T e r m i n o l o g y

We denote by Z, Q, R, and C the ring of rational integers, the fields of rational

numbers, real numbers, and complex numbers, respectively. We put

T={z=C\\z\ = l}.

If p is a rational prime, Zp and Qp denote the ring of p-adic integers and the field

of p-adic numbers, respectively. For an associative ring R with identity element

we denote by Rx the group of all invertible elements of R, by Mn(R) the ring of

all square matrices of size n with entries in R, and by l

n

the identity element of

Mn{R); we put then GLn(R) = Mn{R)x, Rx2 = {a2 | a G Rx}, and

SLn(R) = { a G GLn(R) \ det(a) = 1 }

if R is commutative. We also denote by R™ the set of all (m x n)-matrices with

entries in R; thus Mn(R) = R™, but we use Mn(R) when the ring-structure is in

question. We put R171 = R™ for simplicity. For x G R™ and an ideal a of R

we write x - a if all the entries of x belong to a. (There is a variation of this;

see §18.1.) The transpose, determinant, and trace of a matrix x are denoted by

lx, det(x), and tr(x); we put

x = ( t x ) _ 1

if x is square and invertible. The zero element of R™ is denoted by 0™ or simply

by 0. The size of a zero matrix block written simply 0 should be determined by

the size of adjacent nonzero matrix blocks. If X\, .. . , xr are square matrices,

diag[xi, .. . , xr] denotes the matrix with x\, .. . , xr in the diagonal blocks and 0

in all other blocks. For a complex number or more generally for a complex matrix

a we denote by Re(a), Im(a), and a the real part, the imaginary part, and the

complex conjugate of a. For complex hermitian matrices x and y we write x y

and y x if x — y is positive definite, and x y and y x if x — y is nonnegative.

For r G R we denote by [r] the largest integer r.

Given a set A, the identity map of A onto itself is denoted by id^. To indicate

that a union X = \JieIYi is disjoint, we write X = \_\ieI Y\. We understand that

Yli=a — 1 a n d Yli=a — 0 if ^ /^- For a finite set X we denote by # X or

# ( X ) the number of elements in X. If H is a subgroup of a group G, we put

[G : H] — #(G/H). However we use also the symbol [K : F] for the degree of an

algebraic extension K of a field F. The distinction will be clear from the context.

The idele group of an algebraic number field K is denoted by K£, and for

x G K^ we denote by \X\A, often written simply \x\, the idele norm of x\ see §9.3.

By a Hecke character \ of K, we mean a continuous T-valued character of K^

trivial on K x , and we denote by \* ^ n e ideal character associated with \- As for

the notation concerning localization and adelization of algebraic groups, see §§9.1

and 9.3. We shall employ formal Dirichlet series of the form J^

a

c(a)[a], where a

runs over the integral ideals of the number field in question and [a] is a certain

multiplicative symbol; see §17.10. We shall often put |a| = TV ( a ) - 1 for a fractional

ideal a in both local and global cases.

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