NOTATION
We denote by Z , Q , R , an d C th e ring of rational integers , the fields of rationa l
numbers, rea l numbers , an d comple x numbers , respectively . W e pu t
T =
{*ec|
|* | = i }.
If p i s a rational prime , Z
p
an d Q
p
denot e th e rin g of p-adi c integer s an d th e field
of p-adi c numbers , respectively . Fo r a n associativ e rin g R wit h identit y elemen t
and a n .R-modul e M w e denote b y R x th e grou p o f al l it s invertibl e element s an d
by M™ the i?-modul e o f all m x n-matrices wit h entries in M ; w e put M m = Mf 1
for simplicity . Sometime s a n objec t wit h a superscrip t suc h a s S r o f (2.9.3 ) belo w
is used with a different meaning , bu t th e distinctio n wil l be clear fro m th e context .
For x £ R™ and a n idea l a o f R w e write x - a i f al l th e entrie s o f x belon g t o
a. (Ther e i s a variatio n o f this ; se e §9.1.) Th e transpose , determinant , an d trac e
of a matri x x ar e denote d b y *sc , det(x), an d tr(rr) . Th e zer o elemen t o f R™ is
denoted b y 0™ o r simpl y b y 0 , an d th e identit y elemen t o f R™ b y l
n
o r simpl y b y
1. Th e siz e o f a zer o matri x bloc k writte n simpl y 0 shoul d b e determine d b y th e
size of adjacent nonzer o matri x blocks . W e put GL
n
(R) = (i?™) x, an d
SLn(R) = {ae GL
n
{R) | det(a) = 1 }
if R i s commutative .
If xi, .. . , xr ar e squar e matrices , diag[a?i , .. . , xr] denote s th e matri x wit h
#i, .. . , xr i n th e diagona l block s an d 0 i n al l othe r blocks . W e shal l b e consid -
ering matrice s x wit h entrie s i n a rin g wit h a n anti-automorphis m p (comple x
conjugation, fo r example) , includin g th e identit y map . W e then pu t x* =
f xp',
an d
x = (x*)
- 1
i f x i s squar e an d invertible . Fo r a comple x numbe r o r mor e gener -
ally fo r a comple x matri x a w e denote b y Re(a) , Im(o;) , an d a th e rea l part , th e
imaginary part , an d th e comple x conjugat e o f a. Fo r comple x hermitia n matrice s
x an d y w e write x y an d y x i f x y i s positive definite , an d x y an d
y x i f x y i s nonnegative. Fo r r £ R w e denote by [r] the larges t intege r r .
Given a se t A, th e identit y ma p o f A ont o itsel f i s denote d b y id ^ o r 1^. T o
indicate that a union X = \JieI Yi is disjoint, w e write X = [_\ieI Y{. We understand
that nf= a = * an^ ^2i=a = 0 i f a /? . For a finite se t X w e denot e b y # X o r
#(X ) th e numbe r o f element s i n X. I f H i s a subgrou p o f a grou p G , w e pu t
[G : H] = #(G/H). Howeve r w e us e als o th e symbo l [K : F] fo r th e degre e o f a n
algebraic extensio n K o f a field F. Th e distinctio n wil l be clea r fro m th e context .
As fo r th e notatio n an d terminolog y concernin g Heck e character s o f a numbe r
field and Heck e L-functions associate d wit h them , th e reade r i s referred t o Sectio n
A6 of the Appendix. W e note here only that a Hecke character \ o f a number field
K mean s a continuou s T-value d characte r o f th e idel e grou p o f K trivia l o n K
x,
and x * denote s th e idea l characte r associate d wit h \.
IX
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