For a set S, \S\ denotes the cardinality of S. If 5 is a disjoint union of a family
of sets T\, A G A, we express this fact by S = \J\e&T\. For x G R , [X] denotes the
largest integer which does not exceed x. By R
, we denote the set of all positive
real numbers. For a, b G C, we write a ~ b if b ^ 0 and a/6 is an algebraic number.
By f), we denote the complex upper half plane. For a subset X of R
, we put
dX X \ X°, where denotes the set of interior points of X.
We denote by £(s) the Riemann zeta function. Euler's constant is denoted
by 7. For a function f(x) and a nonnegative integer p,
denotes the p-th
derivative of f(x) with the convention
= f(x). This notation will be used
only in chapter I and appendix II in an obvious context. For a function f(s) which
is meromorphic in a neighborhood of a G C, Res
s = a
/(s) denotes the residue of f(s)
at a.
For a finite group G, G denotes the set of all equivalence classes of irreducible
representations of G on vector spaces over C. For a subgroup H of G and a
representation i\) of H, the induced representation from ip is denoted by Ind^^.
For an associative ring R with unit, Rx denotes the group of all invertible elements
of R. By M(m, n, i?), we denote the set of all m x n-matrices with entries in R. We
abbreviate M(n,n,R) to M(n,R) and set GL(n,R) =
For two fields
Fi and F2 contained in a field K, F\ V F^ denotes the composite field of F\ and F2.
We fix an algebraic closure Q of Q in C. By an algebraic number field, we
understand an algebraic extension of Q of finite degree contained in Q. We denote
by p the complex conjugation.
Let F be a global field, i.e., an algebraic number field or a function field of
dimension 1 with a finite field as the field of constants. For a place v of F, Fv
denotes the completion of F at v. For a G Fv, \a\v denotes the absolute value of
a, i.e., we have d(ax) = |a|vd:r for a Haar measure dx on Fv. By FA and F ^ , we
denote the adele ring and the idele group of F respectively. By F^ (resp. F^) ,
we denote the infinite part of FA (resp. F%) By (FA)/ (resp. (F^)/), we denote
the finite part of FA (resp. F^). For an algebraic group G defined over F, GA
denotes the adelization of G and G^ denotes the infinite part of GA- For a place
v of F, Gv denotes the group of Fv-rational points of G. By a Hecke character of
F ^ , we mean a continuous homomorphism of F ^ into C
which is trivial on F
For x G FA and a place v of F, xv denotes the v-component of x. The finite part
of x is denoted by Xf. For x G F ^ , \X\A denotes the idele norm of x
Let F be an algebraic number field. The ring of integers and the group of units
of F are denoted by O F and Ep respectively. We denote by Ep the group of all
totally positive units of F. The group of roots of unity contained in F is denoted
by WF- We put wp = \Wp\- The regulator, the different, the absolute discriminant
and the class number of F are denoted by Rp- $F, DF and hp respectively. We
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