XV I
INTRODUCTION
where d i s the functio n c i n the presen t case . No w we can find a product A ' of L-
functions o f F an d a product Q of gamma factor s suc h that h!QS ca n be continue d
to a meromorphic functio n o f s o n the whole plane with finitely man y pole s whic h
are al l simple . Clearl y w e ca n sa y th e sam e fo r K'QH. I n th e settin g o f (8) , thi s
A' coincide s with A(s , xi) an d henc e from (8 ) we obtain th e desire d meromorphi c
continuation o f d(s)G(s)Z(s, x) (Theore m 20.5) . I n a simila r way , multiplyin g
by a facto r o f th e typ e A'Q, w e can sho w tha t Z(s, x)E(z, s; /, \) time s suitabl e
gamma factor s ca n b e continue d t o a meromorphi c functio n o n th e whol e plan e
with finitely man y pole s whic h ar e al l simple (Theore m 20.7) .
Strictly speaking, (8 ) is valid only for the character \ whos e archimedean facto r
is consisten t wit h th e weigh t o f / i n a certai n sense . T o obtai n X(s , \)
r a n
arbitrary x , w e have t o replac e £' b y AS' wit h a differentia l operato r A whic h i s
not s o simple.
It shoul d b e note d tha t Garret t gav e i n [Ga ] a formul a fo r th e pullbac k o f th e
standard Eisenstei n serie s o n Sp(r, Z) , fro m whic h on e coul d deriv e a n equalit y o f
type (7 ) i n tha t case . However , h e di d no t carr y ou t th e calculation , whic h wa s
later don e b y Bochere r i n [Bo] . I t ma y b e note d als o tha t equalitie s o f typ e (8 )
were employe d i n a fe w earlie r paper s o f the autho r whe n th e grou p i n questio n i s
obtained fro m a quaternio n algebra .
The final mai n theore m o f the boo k concern s a generalizatio n o f the clas s num -
ber o f a hermitia n form , whic h w e call the mass of G^ relativ e t o a specifie d ope n
subgroup o f G^ . T o explai n th e concept , denot e b y V th e vecto r spac e o f al l n-
dimensional ro w vector s wit h component s i n K o n whic h G ^ act s b y righ t multi -
plication, an d b y x th e maxima l orde r o f K. The n w e can find a finitely generate d
r-submodule M o f V wit h th e propert y tha t xip
txp
r fo r ever y x e M an d M
is maximal amon g such submodules o f V. To make a transparent formulatio n o f the
problem, w e now have to consider the adelization G ^ o f G^, whic h we have avoide d
so far . Takin g a n arbitrar y integra l idea l c i n F , w e defin e a n ope n subgrou p D
of G ^ containin g th e archimedea n facto r o f G ^ suc h that it s f-facto r D
v
fo r eac h
nonarchimedean prim e v o f F i s defined b y
Dv = { a e G% | Mva = M
v
, M
v
(a - 1) C c
v
Mv } .
Then w e ca n find a finite se t B s o tha t G ^ = LLet f G^aD. Le t T b e th e se t o f
elements o f G ^ tha t ac t triviall y o n 3^ ; let F a = G^ n aDa' 1 fo r eac h a e B. W e
then pu t
(9)m(y , c)= Y}
pa n T
ir^ourw,
aeB
where w e understand tha t T G^ an d vol(.T
a\3^)
= 1 if/ ? is totally definite , s o
that w e have
(io) m(p, c) = J2ir
a
•• I]"
1
aeB
for suc h a (p. In thi s specia l cas e clearl y m(£ , c) = #(S ) i f r
a
i s trivial fo r ever y
a, whic h ca n happe n unde r a suitabl e conditio n o n c . Thu s m(p , c) i s simila r t o
the clas s number. Th e poin t o f our formulatio n i s that thi s quantity i s computabl e
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