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 t x, 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* ~ l xp, 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

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