INTRODUCTION
xvn
if (p is anisotropic. Namely , fo r suc h a p w e shall prove, a s the las t mai n resul t o f
this book , a formul a whic h take s th e followin g for m i f n i s odd, th e discrimina t o f
p i s represented b y a unit , an d c = (1) (Theore m 24.4) :
(11)m(y , (1)) = 2
l~%
Y[(k\)
d

Df~n)/2
k = l
n
Yl ^N(V)
k/2D1
F
/2(27r)-kdL(k,
r
k)}
.
k=i
Here t i s the numbe r o f prime ideal s i n F ramifie d i n K, b^ i s an explicitl y give n
constant dependin g onl y o n th e typ e o f 3^ , d = [F : Q], Dp i s th e discriminan t
of F , $ i s th e differen t o f K relativ e t o F, r i s th e Heck e characte r o f F corre -
sponding to K/F, an d L(s, r
k)
i s the L-function o f r
k.
Simila r and somewhat mor e
complicated formula s ca n b e given fo r a n arbitar y c an d als o for eve n n. (Strictl y
speaking, we prove the formula fo r a group D whic h is somewhat differen t fro m th e
above one. ) W e have b^ = 1 if (p is totally definite .
If c = (1), th e quantit y o f (10) i n th e orthogona l cas e i s wha t Siege l called ,
following Eisenstei n an d Minkowski , th e mas s of a genus i n hi s celebrate d theor y
of quadratic forms . Therefor e on e shoul d b e abl e t o deduc e (11 ) fro m hi s formul a
stated i n th e hermitia n case , an d vic e versa , excep t tha t th e deductio n o f on e
formula fro m the other is highly nontrivial. Also , the formulation o f Siegel's formul a
in term s o f th e Tamagaw a numbe r ha s popularize d th e subject , bu t a t th e sam e
time i t ha s obscure d th e significanc e o f othe r classica l problem s i n thi s are a wel l
worthy o f furthe r investigations . Indeed , on e importan t aspec t o f Siegel' s formul a
is that th e mas s i s theoretically computabl e a s h e showe d b y som e examples , bu t
later researcher s completel y neglecte d tha t aspect , apparentl y thinkin g tha t th e
computation i s impracticable i n general .
Without relyin g on the formula o f Siegel's type, we derive (11) as an easy conse-
quence of our methods . T o be exact, w e first prove an equality o f type (8 ) with th e
constant 1 as / fo r eac h class belonging to a fixed genus, and compar e the residue s
of it s bot h sides . Addin g th e result s fo r al l suc h classes , w e obtai n (11 ) wit h n o
ambiguous factors , thu s fulfillin g on e of Siegel's wishes at leas t i n the unitar y case .
Though thi s b y n o mean s supersede s hi s method , i t offer s a ne w insigh t int o th e
nature o f the quantit y m . Indeed , ou r proo f show s tha t m is , up t o som e factors ,
the residu e o f our Eule r produc t fo r / = 1, whic h ma y b e calle d the zeta function
of G^ '. W e believ e als o tha t th e consideratio n o f m(/? , c) wit h a n arbitrar y c i s
natural an d a t leas t technicall y advantageous . Ou r method s ar e applicabl e t o th e
orthogonal case , which th e autho r intend s t o trea t i n a separat e article .
Let u s no w briefl y describ e th e content s o f eac h section . W e first develo p i n
Section 1 a purel y algebrai c theor y o f quadrati c o r hermitia n form s ove r a n invo -
lutorial divisio n algebr a K whic h i s not necessaril y commutative . W e consider G^
as i n (2 ) wit h suc h a K, an d defin e i n Sectio n 2 it s paraboli c subgrou p Pj wit h
respect t o a totall y isotropi c subspac e J o f V. W e the n prov e i n Proposition s 2. 4
and 2. 7 two basic facts o n P^G"/^ x G*) fo r u = diagty;, -p ] as above, which
hold i n a genera l settin g an d whic h pla y crucia l role s i n the proo f o f (7) . Her e P u
is a parabolic subgroup of G u wit h respect t o a maximal totally isotropi c subspace .
In Sectio n 3 we introduce th e notio n o f the denominato r idea l o f a matri x wit h
entries i n the field of quotients o f a principal idea l domain , whic h i s essentially th e
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