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 l x, 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. ix
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