NOTATION AN D TERMINOLOG 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 e C | |z| = l } .

We denote by Q the algebraic closure of Q in C, and for an algebraic number

field K we denote by K"ab the maximum abelian extension of K. 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 and an .R-module M we denote

by

Rx

the group of all its invertible elements and by M™ the i£-module of all

m x n-matrices with entries in M; we put

Mm

— M™ for simplicity. Sometimes an

object with a superscript such as

Gn

in Section 23 is used with a different meaning,

but the distinction will be clear from the context. For x e 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 §1.8.)

The transpose, determinant, and trace of a matrix x are denoted by

tx,

det(x),

and tr(x). The zero element of R™ is denoted by 0™ or simply by 0, and the identity

element of R™ by l

n

or simply by 1. The size of a zero matrix block written simply

0 should be determined by the size of adjacent nonzero matrix blocks. We put

GL

n

(#) = ( i ^ )

x

, a n d

SLn(R) = { a e GLn(R) | det(a) = 1 }

if R is commutative. If xi, ... , xr are square matrices, diag[xi, ... , xr] denotes

the matrix with

1 ? • • • , **•f 1XJL

the diagonal blocks and 0 in all other blocks. We

shall be considering matrices x with entries in a ring with an anti-automorphism

p (complex conjugation, for example), including the identity map. We then put

x* ~

lxp,

and x =

(x*)_1

if x is square and invertible.

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^ or 1^. To

indicate that a union X = (JiG/ Yi is disjoint, we write X = [JieI 5^. We understand

that nf=

Q

= 1

a n d

Ef=a = 0 if a /?. 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

IX