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
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^
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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