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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