XV111
INTRODUCTION
quantity v(a) o f (3) . Mos t noteworth y amon g severa l fact s prove d i n thi s sectio n
are Proposition s 3. 9 an d 3.10 concernin g v(a) fo r a belongin g t o a paraboli c
subgroup o f a genera l linea r grou p an d als o fo r a o f "degenerat e type. " Section s
4 an d 5 concer n quadrati c an d hermitia n form s ove r th e field o f quotient s o f a
ring which i s first a Dedekind domain , late r a principal idea l domain, an d finally a
discrete valuatio n ring . W e introduce th e notio n o f maximal lattices , an d describ e
them i n term s o f a refine d for m o f Witt' s decomposition . W e prov e a produc t
expression o f the typ e G^ PjC wit h th e stabilize r C o f a maxima l lattice .
In Sectio n 6 w e defin e th e spac e 3 ^ an d als o basi c factor s o f automorph y i n
the unitar y case . Variou s elementar y fact s concernin g th e archimedea n versio n o f
Q(f _ p^C an d th e projectio n ma p p : 3^— 3^ ar e collecte d i n this section . Th e
symplectic case , not include d i n these thre e sections , i s treated i n Sectio n 7 .
The adelizatio n G A o f a n algebrai c grou p G an d som e relate d concept s ar e
introduced i n Sectio n 8 . Fo r ou r purpose s i t i s essentia l t o examin e th e cose t
decompositions o f G A relative t o a n ope n subgrou p an d a paraboli c subgroup .
This wil l b e don e i n Sectio n 9 . W e introduc e th e notio n o f automorphi c form s i n
Section 10, prove easy facts o n Hecke operators i n Section 11, and define Eisenstei n
series i n Sectio n 12. W e treat thes e a s function s an d operator s o n 3^ , an d als o a s
objects o n G^ . Ou r expositio n i n thes e section s i s restricte d t o th e unitar y case ,
though w e add som e comments i n th e symplecti c case .
Sections 13 through 15 are devoted to the investigation of a type of local Dirichlet
series that appear s a s an Euler facto r o f a Fourier coefficien t o f an Eisenstei n serie s
on a split group . Thi s local series plays also a crucial role in the computation o f the
Euler factors o f our zeta functions. I n Section 16 we determine th e explicit rationa l
expression for W
p
o f (5) . Th e key fact i n this is Proposition 16.10 , which gives v(a)
for a G Pj. Sectio n 17 concerns severa l formula s fo r th e grou p indice s whic h ar e
necessary fo r th e proof o f (11). W e investigate i n Sections 18 and 19 the Eisenstei n
series o n G
1
whic h w e denote d b y £ i n th e above . W e first giv e a n explici t for m
for eac h Fourier coefficien t an d determin e a product A 7 of //-functions an d anothe r
product Q of gamm a factor s suc h tha t K'Q£ ha s onl y finitely man y pole s o n th e
whole plane. W e then give an explicit formul a fo r the residue at a special pole when
the weigh t i s 0.
We state our main theorems onZ(s, \) an d E(z, s, / , \)
m
Sectio n 20, and prove
them i n the nex t thre e sections . On e o f the mai n technica l difficultie s arise s i n th e
analysis of the pullback denoted by H(z, w; s) in the above. Thoug h the descriptio n
of P u; \G a; /(G^ x G^ ) give n i n Sectio n 2 is not complicated , w e have to describ e i t
in connection with various open subgroups of the adelized groups. I t shoul d als o be
mentioned tha t i n order t o obtain (7 ) an d (8) , we must choos e £' carefully , becaus e
an arbitrar y o r a seemingly natura l choic e of £' ofte n produce s a vanishing integra l
or ambiguou s factors . I t i s on e o f th e mai n point s o f ou r treatmen t t o giv e suc h
formulas i n nonvanishin g exac t form s wit h al l factor s explicitl y determined .
As we said earlier , t o establis h th e meromorphi c continuatio n o f Z(s, \)
m
th e
most genera l case , i t i s necessar y t o appl y a certai n differentia l operato r t o £. I n
Section 23 we define such an operator an d prove a formula o n its effect o n each ter m
of £, whic h eventuall y lead s t o th e proo f o f the desire d fact . Finall y i n Sectio n 2 4
we prove a formula fo r m(p , c) when ip is anisotropic. I f ip is not totall y definite ,
we can deriv e fro m i t anothe r formul a concernin g vol(r
a
\3^) (Theore m 24.7) .
The Appendix at the end of the book consisting of eight sections contains various
facts whic h coul d hav e been include d i n the mai n text , bu t woul d hav e interrupte d
Previous Page Next Page