NOTATION AND TERMINOLOGY

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

n

, we put

dX — X \ X°, where X° 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,

f(p\x)

denotes the p-th

derivative of f(x) with the convention

f(^\x)

= 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) =

M(n,R)x.

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

x

which is trivial on F

x

.

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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