x NOTATION AND TERMINOLOG Y

denote by 1(F) the ideal group of F. For an integral ideal f of F, If(F) denotes the

ideal group of F modulo f, i.e., the group of all fractional ideals relatively prime to

f. For x £ F£, div(ar) denotes the fractional ideal Y\v

Pvv

where v extends over all

finite places of F, pv is the prime ideal corresponding to v and av = ordvxv with

the normalized additive valuation ordv at v. By ri(F) and 7* 2 (F), we denote the

number of real places and of imaginary places of F respectively. We denote by Jp

the set of all isomorphisms of F into C and by Ip the free abelian group generated

by Jp. For a G F, a 0 means that a is totally positive. We denote the maximal

abelian extension of F in Q by Fab. For a £ F%, [a,F] £ Gal(Fa&/F) denotes the

image of a under the Artin map. For an abelian extension L of F of finite degree

and a fractional ideal a of F, which is relatively prime to the conductor of L, (—j—)

denotes the Artin symbol.

Let K be an algebraic number field which is an extension of F of finite degree.

By ResK/p, we denote the restriction homomorphism from IK to Ip. By Inf^/F?

we denote the homomorphism from Ip to IK such that, for a £ Jp, Inf^/F^) is the

sum of all elements of JK whose restrictions to F coincide with a. The norm map

and the trace map from K to F are denoted by

NK/F

and by

TTK/F

respectively.

The relative discriminant (resp. the relative different) of K over F is denoted by

D(K/F) (resp. *K/F).

By a CM-field, we understand a totally imaginary quadratic extension of a

totally real algebraic number field. For a CM-filed K, 3 £ IK is called a CM-type

of K if $ = Y^Ji=i

ai

£ IK with Oi € JK and if $ + $p is the sum of all elements in

JK- We often identify $ with the set of isomorphisms {Ji, 02, • • • , crn} or with the

representation of K into M(n, C) sending a £ K to the diagonal matrix with a

a i

,

a0"2,

...,

aan

on the diagonal entries. A point z = (21,22,... , zn) G

ion

is called

a CM-point if for a CM-field K, a CM-type $ = {cr1? cr2,... , crn} and for a e K,

Zi—aa\ 1 ^ i ^ n holds.