XI V
INTRODUCTION
sense. Bu t t o avoi d excessiv e details , w e d o no t giv e her e th e precis e definitio n i n
the genera l case. Thoug h thi s ma y no t b e the bes t wa y to produc e a zeta functio n
in othe r cases , i t ha s th e advantag e tha t w e ca n connec t th e Heck e eigenvalue s
naturally an d directl y t o th e desire d Eule r produc t fo r a larg e clas s o f classica l
groups. Also , in certain case s this definitio n allow s us to express our Eule r produc t
in terms o f the Fourie r coefficient s o f / , thoug h w e do not touc h o n that aspec t i n
this book .
For th e purpos e o f illustration , le t n = 2r , G^ = Sp(r, Q) , an d F = Sp(r, Z) ;
then w e tak e E G? an d v{a) t o b e th e produc t o f th e denominator s o f th e
elementary divisor s o f a . I f r = 1, the n th e grou p i s SX2(Q ) an d th e serie s ha s
the for m
oo
(4) T(s ) = J2 A(diag[m- \ m])m~ s,
7 7 2 = 1
which produce s a n Eule r produc t o f degree 3 , that i s usually calle d th e symmetri c
square zet a functio n associate d t o a series o f type (1), but i t i s also natural t o cal l
it a n Eule r produc t o n 51/2 (Q), wit h n o reference t o GL2(Q) .
Now ou r first mai n tas k i s t o sho w tha t thi s typ e o f serie s wit h v define d i n a
similar manne r ha s a n expressio n
(5) A(s)%(s) = Y[w p(N(p)-°y\
p
where A is a produc t o f L-function s o f F , p run s ove r al l th e prim e ideal s i n F ,
and W
p
i s a polynomial wit h constan t ter m 1 whose degree i s n[K : F] fo r almos t
all p , excep t whe n G^ i s a symplectic group or an orthogonal grou p of odd degree ,
in whic h cas e th e degre e o f W
p
i s n + 1 o r n 1 accordingly . Thi s fac t i s purel y
algebraic, or rather local , and can be formulated withou t th e notion of automorphi c
forms. Therefor e w e obtai n th e resul t fo r a n arbitrar y G^ o f typ e (2 ) (Theore m
16.16).
Now, assumin g tha t K ^ F, tak e a n arbitrar y Heck e characte r x o f X an d
denote b y Xi it s restriction t o F ; denot e als o by T(s, x)
a n
d Z(s, x) th e twists of
T an d th e right-han d sid e of (5 ) b y x b y viewing the m a s Euler product s ove r K\
further denot e b y A(s , xi) th e twis t o f A by xi - The n
(6) A(a , xi)£(s , x ) = Z(s, x) -
Our nex t proble m i s to prov e tha t i f / i s a holomorphi c cus p for m i n th e unitar y
case, then Z(s, x ) times suitable gamma factors ca n be continued to a meromorphi c
function o n th e whol e plan e wit h finitely man y possibl e simpl e poles . I n principl e
our method s ar e applicabl e t o nonholomorphi c forms , bu t th e incomplet e stat e o f
knowledge o f suc h form s make s i t difficul t t o find explici t form s o f th e gamm a
factors. I n th e holomorphi c case , however , i t i s relativel y eas y t o calculat e th e
gamma factors , whic h i s th e mai n reaso n wh y w e restric t ou r expositio n i n late r
sections t o holomorphi c forms .
We prove the desired result o n Z fo r G^ whe n K i s a totally imaginary quadrati c
extension of totally real F, in which case G^ i s a unitary group acting on a hermitian
symmetric spac e 3^ - Ther e ar e severa l reason s wh y w e conside r unitar y group s
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