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l
and dir@,
(2,. . . , 2)]s+hlIdim[(2,...,2)]
if I’ is commensurable with a Hilbert modular group. One can show by means of the residue theorem that in the second case we also have an equality (n = 1). This gives us
$4 An Algebraic
121
Geometric Method
4.7s Corollary. If l? is commensurable with a Hdbert x(M2)
= (l)n
modular group, we have
+ dim[I’, (2,. . . ,2)]  E:,
where
1 { 0
‘=
ifn=l ifn>l.
The arithmetic genus of 2 is defined as g := x(0,)
= arithmetic genus.
By duality we have
x(Q) = &qpg, = (l)“x(&) > p=o
where gp = hpto = dimR$(z)
.
We obtain 4.74
Corollary.
The arithmetic
genus is given by
g = 1 + (l)n
dim[I’(2,. . . ,2)]s .
The Final Formulae. Let a E Hn be an elliptic fixed point of l?. We define the contribution E(I’, u) by
where C = (cl,...,&)
are the rotation factors belonging to A4 (“A4 # id” means that the transformation induced by M is not the identity, i.e. M # (fE,
If K.is a cusp, the contribution Shin&u Lseries.
. . . , fE)
.)
L(I’, K) has been defined in $3 as a certain
4.8 Theorem. Let r c SL(2,R)” be a discrete subgroup such that the extended quotient (H”)*/l? is compact. we asnme that the first irreducibility
Chapter
122
assumption reducibility 1 + (1)”
I.219 is satisfied, assumption I,,$.lZ. dim[I’,
(2..
II.
Dimension
Formulae
and, if Hn/l? is compact, also the second Th en the following formula holds
. ,2>]ll = (l)nvol(H”/I’)
+ c
qr,
a> + c
qr,
ir
tc) )
K
a
where a (resp. IC) runs over a complete set of representations of requivalence classes of elliptic fixed points (reap. cusps). This expression also equals the arithmetic genus g of a desingularization of X. A Simple Special Case. Assume that n > 1 is an odd number without elliptic fixed points. Then the formula simplifies g = 1  dim[I’,
(2..
. ,2)]s
and that
I’ is
= vol(Hn/I’)
So the genus is a negative number. As a consequence we have for example that the field of automorphic functions is never a rational function field under this assumption, because the genus would otherwise be 1.
$5 Numerical
Examples
in Special
Cases
The numerical evaluation of the invariants occuring in Shimizu’s function fi (r) is in general very complicated. In some cases it can be calculated explicitly. We collect some wellknown results, mostly without proofs. For more details we refer to Hirsebruch’s paper [30].
The “main of a fundamental
term” of Shimizu’s domain of l? with
function respect
comes from the volume to the invariant measure
vol(I’)
dw=& g2 dv = Euclidean
measure
= dzl
dyl . . . dx,
dy,, .
This volume has been calculated by Siegel [59] in the case of the Hilbert modular group. To be more precise, he expressed it in terms of the Dedekind Cfunction. Let K be a totally real number field of degree n. Its Dedekind Cfunction is defined as
k(s) = C(Na))” , where the sum is extended over all integral ideals a (0 # a C 0). This series converges if the real part of s is greater than 1 and defines an analytic function in this halfplane. It has an analytic continuation as a meromorphic function into the whole splane with a single pole (of first order) at s = 1.
$5 Numerical
The
Examples
in Special
123
Cases
function
is invariant
under
5.1 Proposition. group
(dK
= d,“‘2~““/2r(s/2)nCK(s)
k(s)
with
respect
the transformation
s + 1  s. Siegel’s
The volume of a fundamental to the invariant measure
d&J = (4+“$
= discriminant
domain
(dv = Euclidean
of K)
result
is [59]:
of Halbert’s
volume
modular
element)
is vol(SL(2,o))
By means
of the functional
= 27l)Q(l) .
equation
(1)%(l) The
trivial
we obtain
= d$22“r2nCK(2).
estimation
gives us 5.2 Corollary. vol(SL(2,o))
Explicit
formulae
for CK( 1)
> 212nr2nd$2
are known
.
in the case of a real quadratic
field
K = Q(h), where
a > 1 is a squarefree
natural
number.
The discriminant
of K is given
by d=4a d= For the following given.
result
a
ifar2,3 ifar
1
mod4 mod4.
we refer to [30], p.192,
where
5.3 Proposition. Let
K = Q(h)
(a > 1, squarefree)
further
references
are
Chapter
124
be a real quadratic field.
Then
II.
Dimension
Formulae
for a G 1 mod 4
andforaE2,3mod4
CK(1)= &m(a) + 2.
ul(a  b2)).
c
l 1 be a squarefree natural number a>5,
(a,S)=l.
of order 2 and 3 GOT the Hilbert modular e Teal quadratic field K = Q(G). Their number AZ, As ia given by the following formulae There
e&at only
QTOUP
r
=
elliptic
%(2,0)
fixed points
Of th
1)
azlmod4:
Az = h(4a) ,
2)
aG3mod8:
Aa = lOh(a)
3)
aG7mod8:
Az = 4h(a),
A3 = h(3a) ,
Al = h(12a)
fi(r)
We restrict
= dim[I’,
.
At = h(12a)
In all three cases the fized point8 of order 3 (1,1) together with one of type (1,2)) *. We next determine the contribution and 3 to the rank formula
. . in pairs (one of type
occur
of the elliptic fixed points of order 2
(2r,.
. . ,2r)],
T>l.
to the case
rGOmod6, because this is sufficient, to determine the arithmetic dim[l?, (2,. . . ,2)]).
genus (equivalently
1) e = order (a) = 2 The contribution to the rank formula is
WW 2)
(=
E2(r,aN
=
&
+
,,‘(,
+
1)
=
i.
e = order (a) = 3 *
This is different if one also considers d divisible by 3 (see for example [30], p.237).
126
Chapter
((el, e2)
Type 1
Let C be a third
Type II
e2)
1 = 3
Dimension
Formulae
:
root of unity. The contribution
((el, Jfw+)
= (Ll))
II.
= (1,2))
is
:
1 1 (1 [)(I (2) + (1 C”)(l (
 .
5.5 Lemma.
Let I? c SL(2, R)2 b e a discrete irreducible subgroup such that the extended quotient (H2)*/l? is compact. The contribution of the fixed points of order 2 and 3 to the rank formula for dim[I’, (2r,. . . ,2r)], r 3 0 mod 6, is given by 1)
e = order (u) = 2
qr,q 2)
= i/8.
e = order(u) = 3 Type 1
((el,
e2)
= (1, 1))
qr, Type II
((el,e2)
U) =
i/9
= (L2))
E(r,q
=2/9
The Contribution of the Cusps. We recall that the cusp classes of the Hilbert modular group l? = SL(2, o) are in llcorrespondence with the ideal classes. If 6,:; U,CEO C
is a cusp, then a := (a, c) represents the corresponding ideal class. To transform K to 00 we choose a matrix adbc=l; which has the property ACO=K.
b,d E al
$5 Numerical
A simple
Examples
calculation
QP ; K 7 6>
in Special
shows
Cases
that
127
the conjugate
oC5&=l;
group
ArI’A
equals
cvEo,6Eo,j3Ea2,yEa2
The translation module of this the group of unit squares:
>
is am2, and the group
group
.z unit
A = {E2, So in the case n > 1 the contribution given by
.
of multipliers
is
in 0) .
of the cusps to the rank
formula
is
where d(a) denotes the discriminant of a given ideal a which, in the totally real case, is nothing else but the square of the volume vol(a) (see App. I for the definition of d(a)). We recall that this expression does not change if one replaces equivalent ideal. It vanishes, of course, if there exists a unit E E o* with We hence
make
a by an
Ne = 1.
the
Assumption. Each
unit
E in o has positive
Under this assumption sgn Na merely generated by a. We hence may define $((u)) on the group of all principal ideals. by a totally positive element.
norm
(NE = +l).
depends upon the character
the principal
It is 1 if the principal
ideal
?it is the subgroup of the group of all principal consists of ideals generated by a totally positive element. so is defined on the factor group +
is generated
ideals, which character
OUT
(1, 1).
We denote group Z/7i
by Z the group of all ideals of K. The are the ideal classes and those of I/‘?& ideal classes. Both groups are finite. We have n/7&
(a)
:= sgn Na
Notation:
1c, := tipit
ideal
c 2/7d
.
elements of the factor the socalled narrow
128
Chapter
Because Z/Nt
II.
Dimension
Formulae
is a finite abelian group, we may extend 1c,to a character x : z/76
t s1 = {C E c, \(I=
1).
We have proved: 5.6 Lemma. Assume that each unit has positive norm. Then there esists a character x on the group Z of all ideals depending only on the narrow ideal class and satisfying x((a)) = sgn Na .
It is worthwhile the case if
asking whether 2/7h
especially
x can be taken to be real. This is, of course, N z/x
x 7l/7& )
if the order of Z/7t, i.e. the class number
h of K, is odd.
Using a character x described in Lemma 5.6 we may rewrite the contribution L(I’, 6) as follows: We recall that A is the group of all squares of units. This group is a subgroup of the group of all units of index 2n. This gives us
c sgn, aa/,. INal
c 57!$=2n o:a=/A
because all units have positive norm by assumption. If a runs over a complete system of representatives of aS2/o*, then .a2 = x runs over all integral ideals in the class of a2. This gives
c
E
= N(a2)X(a2)
F
f$$
,
a=/o* where x runs over all integral ideals in the ideal class of a2. Here the Smite sum means the limit value of the Lseries
x(x) c Jwa
which
converges
for s > 1 at s = 1. The expression
does not depend on the ideal b and hence equals the discriminant field K. We obtain for the contribution of our cusp $dg2n(a2)
F
$f$
.
do of the
85 Numerical
Examples
in Special
129
Cases
If we make the further assumption that x is real, we have x(a2) and obtain for the sum of the contributions of all cusps $g2L(l, where
= x(a)2
= 1
x) )
L(l, x) denotes the limit value of the Lseries L(s, xl = C
x(W(a)”
,
SK0
where
a runs over all integral
ideals.
5.7 Proposition. Assume that the norm of each unit is positive. Assume fwthermore that the character in 5.6 can be chosen to be real (for example if the class number is odd). Then the contribution of all cusps to the rank: formula is
It is a wellknown fact that this expression is always unequal zero. We finally consider a very interesting special case, namely the case of a real quadratic field K = &(fi), P prime The following beautiful formulae can be found in Hirzebruch’s paper [30], 3.10, where further comments and references are given: In Q(,/jS) a unit of negative norm exists if and only if p=2
or
pElmod4.
Hence precisely in this case the contribution of the cusps to the rank formula is 0. In the case p s 3 mod 4 the squares of ideals generate a subgroup of index 2 in the narrow class group Z/7&. This implies that there is exactly one real character x with the properties in Lemma 5.6. By means of the decomposition law of the field Q(Jii) one obtains L(% x) = where the L, are the usual Dir&let formula gives
J54W&)
Lfunctions. The socalled class number
L4(1) = ?4l&(4) L,(l) where h(4)
?
= Fp1/2h(p)
(=class number of Q(i)) is one.
Chapter
II.
Dimension
for the arithmetic
genus
130
We now have the complete
formula
Formulae
g=l+dim[I’,(2,...,2)]0 =l+dim[I’,(2,...,2)]h in the case K = Q(d), p prime, p > 5. In the remaining cases p = 2, 3, 5 the determination of the elliptic fixed points is also continued in the mentioned paper of Prestel [52]. (Actually those three special cases have been treated by Gundlach in an earlier paper [21].) 5.8 Theorem. Let p be a prime,
K = Q(&.
The arithmetic
genus
g=l+dim[P,(2,...,2)]0 =l+dim[I’,(2,...,2)]h of the Hdbert modular surface with respect l? = SL(2, o) is given by the formulae
to the Hdbert
modular group
for p = 2,3,5,
g=l g = &+l)
+ !!p
+ qd
for p 3 1 mod 4,
g = &(1)
+ ;h(p)
+la(i2p)
for p z 3 mod 8,
g = &(1)
+h(;2p)
p > 5,
for p 3 7 mod 8.
The Group I?. We define
r := ivriV , NI=(;
;),Nz=(;
Tl).
The action of P on HZ is equivalent to the action of I’ on the product of an upper and a lower halfplane (by means of the usual formulae). Consider an element a E K with a(r) > 0 ,
a(*) < 0 .
The matrix
has obviously the following property: The groups AI’Al
and
l?
$5 axe
Numerical
Examples
in Special
If there is a
commensurable.
131
Cases unit
a with
the above property,
both groups
axe equal. The structure
of the elliptic fixed points of I? is dual with
that of l?.
The numbers of classes of fixed points of a certain order are equal, but the types are changed. In case of fixed points of order 3 precisely the But in the cases which we two types ((1,l) and (1,2)) are interchanged. considered the elliptic fixed points of order 3 occur in pairs (5.4). So in the formulae for the arithmetic genus g of I and g of r only the contributions of the cusps may differ. It is obvious that they precisely differ by a sign. This gives us the following beautiful result of Hirzebruch. (For sake of completeness we also include the special cases p = 2 and 3). 5.9 Theorem.
Let p be a prime. 9  g = dim[I’, = dim[I’,
The difference
of the arithmetic
(2,. . . ,2)]e  dim[I’, (2,. . . ,2)]  dim[I’,
genera
(2,. . . ,2)]s (2,. . . ,2)]
equals 0
ifp=2 OT~OT pEl if P > 3 andpE3
h(P)
mod4, mod4.
This result implies for example, that the field of modular functions with respect to l? is not a rational function field if p > 3. Pinal Remark. One might conjecture that the main term in the formula metic genus is the term which comes from the volume of the fundamental following sense. Let rm c SL(2, R)” (n = n(m) may vary with m) be a sequence of groups commensurable modular group, such that for the different m,m’ with n(m) = n(m)) the T,! are not conjugate in SL(2, R)“. One then might conjecture vol(Hn/rm)+oo
1)
2)
g(r,)

(wOwwm)
vol(Hn/r,)
4.8 for the arithdomain in the
with a Hilbert groups rm and
ifmrco ~
o
ifm+oo.
This conjecture would imply that only finitely many conjugacy classes of groups r with rational function field exist. Wellknown estimates for the class number of an imaginary quadratic field and the estimate 5.2 show that this conjecture is true for the sequence of the usual Hilbert modular groups of K = Q(@),p p rime (a. 5.8 for the formula of the arithmetic genus). The conjecture is unsettled even if one restricts to a fixed n > 2 and to a usual Hilbert modular group. For the case n = 1 see [60].
Chapter III. The Cohomology Hilbert Modular Group
of the
$1 The Hodge Numbers of a Discrete Subgroup r C sL( 2, W)n in the Cocompact Case In this
section
we compute
where
l? c
where
I has no elliptic
llxed
simple
invariant,
the volume
SL(2,R)
the Hodge
n is a discrete
namely
numbers
subgroup
points
with
all those
compact
numbers
of a fundamental
quotient
Hn/I’.
can be expressed domain
with
respect
In the
by means
case of a
to the invariant
measure. The
results
of this
We consider
section
open
are due to Matsushima
domains
D, c C
a,...,
of the complex function
plane
equipped
with
hi : D; t We may consider
and Shimura.
the “product
a Hermitean
metric,
i.e. a positive
R+ = {CTE R 1I > 0). metric”
0
h= on the domain
D=D1x...xD,. Via
the identification C” ( 21 ,...,Gl)
(21,y1,...
R2” ,Zn,Yn
>
C”
Chapter III. The Cohomology of the Hilbert Modular
134 the associate
Riemannian
metric
is given
Group
by
h hl *.
9= 0
(See App. III, Sects. IXXI.) Such a metric Sect. XII). We especially have the relation
has the K%hler
property
(A III,
A=20=2Ei. We make
use of this
relation
to prove
1.1 Lemma. If a, b are subsets
A( with
and if f is a P’function
of (1,. . . , n} D (= D1 x . . . x D,), we have
our domain
a certain function
f dz, A dI&,)=
gdz,
A d&,
g.
(Recall: dz, = dz,,
A . . . A d.zap ,
where a = {al,.
..,c+},
15~1
= 2SijY:
< dzi,dzj
> =
=O
letters.) is used in
’
’
In general < qu, is defined
as the determinant
b, c), qi?i, a, 2) >
of a certain
m
x
mmatrix
(m = a! + p + 27).
If (6, a, E) # (b, a, c) this matrix contains less then m nonzero components. Its determinant therefore is zero. In the remaining case we obtain 0 yz . yi . det
The
2E(")
2E@) 0 0
star operator
0 0
0
0 0 0
0 0
2E(7) 0
2E(7)
is defined
= (1)
. y; .
d+72Q+@+z7y;
by the formula
wQ=wAw’. With
the information < CR(a,
Both
we obtained b, d), i2(& i,ci)
sides are zero except
We now
case both
the pairing
> ‘WO = qu,
b, c) A R(ii, iG,2) .
= (b,u,d),
sides are (ycyd)”
up to constant
obtain
1.5 Lemma.
it is easy to verify
when (i?,~,~)
and in the latter
about
Let w be a differential
form
w = fQ(u,
of the type b, c) .
factors.
cl
138
Chapter
We have = b) Z%J = c) a(*w) d) a(*,)
a) &
0H 0 H = 0 = 0
III.
The
Cohomology
Modular
Group
f is antiholomorphic in the variables zj, j E b U d. f ia holomorphic in the variables zj, j E a U d. in the variables zj,j E b U c. * f is antiholomorphic in the variables zj, j E a U c. * f is holomorphie
1.51 Corollary.
The relations
in the variables coming from a, an
are equivalent with: w is holomorphic tiholomorphic in the variables coming variables coming from c U d. (A function p(z) of one complex P(Z) is holomorphic.) Proof.
of the Hilbert
from
variable
b and
is called
locally
constant
antiholomorphic
in the
if z H
We have dw=dfr\SZ(a,b,c)=O
iff Gaf
=Oforj
E bud.
This means  by the CauchyRiemann equations  that f is antiholomorphic in the zj’s (j E b U d). This proves a) and similarly b). For c), d) one has to use 1.4. 0 The
corollary
Cancellation
1.51 implies Rule.
If
a certain
w = fn(a,
cancellation
b, c) satisfies
rule. the equation
au=&=a~w=8+w=o, then the same is true
of
instead of w and conversely.
phic
We now want to transform “antiholomorphic variables” ones. For this purpose we consider the diffeomorphism bb
: H” 
H”
Z
W = 0(,(Z),
zj
for j # b
Zj
for j E b.
where Wj
=
into
holomor
$1 The
The
Hodge
Numbers
of a Discrete
139
Subgroup
function
is holomorphic in all variables if fn(u, b, c) satisfies the condition 1.5. What does I’invariance of w = fR(a, b, c ) mean for the transformed g(z)
To express
this we introduce
=
a)d)
in
function
f(ObZ)?
the notation
(: ii>=(2 ib)=(ii :1)(: :) (i !l). M I+ M
Obviously
defines
an automorphism
M = (Ml,...
of SL(2,
, Mn) E SW,
R). If
R)”,
we define
N=Mb Nj = The
by
Mj
ifjgb,
Mj
if j E b.
groups
rb = {Mb 1M E I?} c SL(2,R)” satisfy the same assumptions general; the quotients Hn/r they carry different “analytic The
of w = fS2(a, b, c) means
rinvariance
I
=
1.2 as l? (but they are different from and H”/lTb are topologically equivalent structures”).
+ dj)2 n(cj~j
n(cjrj
jEa because
+ dj)2f(~)
for M E I’,
jEb
the forms Wi
=
dzi A ai Yi!
are invariant. For the function g(z)
=
f(abz)
we obtain
g(Mz)
=
n jEaUb
PJ
se a;l(Mabz)
= Mbz.)
(cjzj + dj)2g(z) for M E lYb.
l? in but
140
Chapter
III.
The
Cohomology
of the Hilbert
Modular
Group
Holomorphic functions g with this transformation property are special examples of automorphic forms as considered in Chap. I. Assuming a certain condition of irreducibility (1.4.12) we were able to show (1.4.13) that these functions vanish unless or
aUb=0
aUb=
{l,...,n}.
In the first case we have the constant functions, in the second case automorphic forms of weight (2,. . . ,2). We now obtain the complete picture of the Hodge spaces ‘FPq(I’). Th ere are two possibilities for a nonvanishing harmonic form w = fcqu, b, c) .
Case 1. a U b = 0. In this case we have w = const . wC .
These forms are actually invariant  not only with respect to our discrete subgroup  but with respect to the whole group SL(2, R)“, and they are harmonic as follows from 1.4 (and aR(u, b, c) = ~Q(u, b, c) = 0). We collect these “universally invariant” forms in the socalled universal part of the Hodge spaces ifp=q(andO is an automorphic form of weight (2,. . . ,2) with respect to lYb. 1.6 Theorem. Let I?
c SL(2,R)n b e a discrete subgroup which satisfies the assumption 1.2 (especially, that Hn/lJ is compact) and the iweducibdity condition I.4.1,% We have 1) in the case p + q # n
$1
The Hodge Numbers of a Discrete Subgroup
141
2) in the case p + q = n TP(r)
[lTb,(2,. . . ,2)]. $ bc{l ,...,nl #b=q
E 7g”
The dimensions of the spaces [rb, (2,. . . ,2)] have been computed in Chap. II by means of the Selberg trace formula in connection with an algebraic geometric method (to come down to the “border weight” (2,. . . ,2)). 1.61 Corollary. If ( in addition) I’ has no elliptic fixed points, of the spaces [rb, (2,. . . ,2)] do not depend on b. We have dim[rb,
(2,. . . ,2)] = vol(Hn/r)
+ (1>n+r
where the volume is taken with respect to the invariant As a special case of 1.6 we obtain
the dimensions )
volume
element.
the spaces
v(r) = {wEAfIg
1Aw
= O}
of all harmonic I’invariant differential forms of degree m. Notice. These spaces do not depend on the holomorphic structure on the underlying Riemannian metric. From the equation A = 20 we know that A is compatible (p, q)bigraduation, hence
but only with
v(r) = $ w(r). p+q=m
The dimensions
of these spaces are denoted by b” = dimtim =
c p+q=m
hpfq .
They are 0 if m > 2n (or m < 0). 1.7 Theorem. Under the assumptions 1) in the case m # n bm =
of 1.6 and its corollary
we have
( mT2 > if m is even, 0
if m is odd.
2) in the case m = n (&)
if n is even,
0
if n is odd.
b” = 2” . dim[I?, (2,. . . ,2)] +
the
142
1.71
Chapter
Corollary.
III.
The
Cohomology
of the Hilbert
The alternating sum of all the b” F(l)jti
= (2)n
. vol(H”/I’)
Modular
Group
is
.
j=O
Final Remark. groups. From
The numbers calculated the general Hodge theory
H”(r)
above are actually (App. III) follows
E Hm(Hn/l?,
dimensions
of cohomology
C)
(singular cohomology with coefficients C). It should be mentioned that the last formula (corollary of 1.7) is also a consequence of the Gaul3Bonnet formula which expresses the Euler characteristic (= alternating sum of Betti numbers) by means of the curvature and the volume. If l? has no elliptic fixed points, one furthermore has
where
SF’ denotes
the sheaf
of holomorphic
$2 The Cohomology of a Cusp
pforms
Group
on the analytic
manifold
Hn/r.
of the Stabilizer
Let
D c R” be an open We assume finite index
domain and I? a group of C”diffeomorphisms of D onto that I? acts discontinuously and that I? has a subgroup which acts freely on D. We denote by
itself. ITo of
wL(w C” differential the linear space of all l?invariant We may consider the socalled “de Rham complex” ... 
ML(D)r
d
Mg’(D)r
(“complex” means d. d = 0) and the de Rham actually Cvector spaces) HP(r)
= HP((DJ))
forms
+
cohomology
= CP/BP )
of degree
p on D.
... groups
(they
are
$2 The
Cohomology
Group
of the Stabilizer
143
of a Cusp
where Cp = ker(M&(D)r BP = im( ML1 Of course
+ ( D)r
M~‘(II)~) +
M&p)r).
we have Hp((D,r))=O
if
Notice. By the theorem of de Rham and the singular cohomology group convex) we have furthermore
there
pn.
is a natural isomorphism C). If D is contractible
HP(D/I’,
HP((D, I?)) S Hp(D/r,
H*(I’,
C) denotes
the group
cohomology
I’)) D is
C)
E HP(r,c) where
between HP((D, (for example if
’
of I? acting
trivially
on C.
We now assume that a discrete subgroup r c SL(2, R)” with cusp 00 is given. We want to compute the cohomology of the stabilizer roe. Recall that the stabilizer roe consists of transformations of the form
We have two types of differential forms all thEse transformations, namely 1) dq
A..
which
are closed
and invariant
under
. A dz,,
2) *+y.../\f+ Y,P
where a = (c81,...,upL
l<ar
1
a3
j=l
where ~1, . . . , Kh denotes a system gruence group of a Hilbert modular which maps isomorphically to the theory is due to Harder [26].
of representatives of the cusps. In the case of a congroup we will construct a certain subspace of H”(I) image of by means of Eisenstein series. This
H"'(r)
We start with the basis of H”(I’,) Let w be one of the basis elements, i.e.
1) 2)
w=
w = 4/a
&a 7 Ya
h dxl A..
which has been constructed in 52.
(m
Ya
where aUb= are r,invariant
{l,...,
anb=B,
n},
and closed. Their
classes
#a=mn, define
PTOO~.We show that the differential forms dz, A CE’i,
A dzb
Ya
and *Adx Ya
1 A ...
A dx,
a basis of H”(I’,).
149
$3 Eisenstein Cohomology
are cohomologous upto a constant factor. This obviously follows from the fact that a differential form of the type $
A dxb A dy, , a # {l)...)
n},
bUc=
(1)“‘)
n},
defines the zero class if c # 0. This is trivial if c n a # 0. But if c n a = differential form yi* 2
is I’,invariant,
AdXb
Adyct
,
0, we choose some index i E c. The where c’ = c  {i} ,
and its exterior derivative is up to a sign ha yc A dxb A dy, .
Let now w be one of our basis elements, i.e. Case 1: w = (dy,)/y,
,
&se2:~=~“~~~~Adzb
(aUb={l,...,
n},aC{l,...,
nl}).
These forms are I’,invariant, but we want to have Iinvariant It looks natural to construct them by symmetrization: E(w) :=
forms.
c w 1M. MEr, \I’
We use the notation wIM:=M*w. We have (wIM)IN=wlMN, and hence w I M does not depend on the choice of the representative M. The main problem will be the question of convergence. The formulae dz 1M = (cz +d)2dz, &I
M = (cz+d)2&,
y 1M = I cz + d I’ Y , dy = (1/2;)(dz
 do)
show that the series E(w) can be expressed by Eisenstein series of the following type:
Chapter
150
We consider
is independent the Eisenstein
of j and that series = Ez,p(z)
c MEr,\r
considered E(w)
where p(z) is a linear More precisely
For the total
Cohomology
r is integral.
=
I, §5 we already
The
(Y, ,L?E Z” of integers.
two vectors
&J(Z)
In Chap. ma&)
III.
combination
of the Hilbert
We assume
We then may
qcz
Modular
that
(formally)
+ d)“N(cZ
consider
+ d)+
the case ,f3 = 0. We now
Group
.
obtain
(for
= P(Z) . WY of the Eisenstein
series introduced
before.
weights
we obtain: Case 1:
r=o
(w=~Ca+q,
Case 2:
r = 1
(w = c c, dZ$fia
From our assumption second case
A dza).
a c { 1, . . . , n  1) we furthermore
obtain
in the
a#P. Up to now our consideration the question of convergence.
has been formal. We now have We have already proved that
to deal
with
c IJqcz +d)I2r converges for all real r > 1. The same proof shows that this series does not converge if r = 1. In the first case (2~ = oj + /3j = 0) we are rather away from the border of convergence. The second case looks better. Here we are precisely at the border of absolute convergence (r = 1). Following an idea of Hecke we can define E,,p(z) in this case as follows: We first introduce for real s > 0 the series E,,p(z,
This
s) :=
series converges
c r,\r
N(CZ + fpN(cF
absolutely
+ cl)@ 1 N(cz
+ d) 12s .
(in the case 2r = oj + pi = 2, s > 0).
$3 Eisenstein
One may
exists.
151
Cohomology
ask whether
If this happens
The series fined by
E,,~(z)
the limit
we say: admits
He&e
summation.The
value
of this
series
is de
We keep the notation E,,&) where
the symbol
“=”
“ = ” c
N(cz
indicates
that
+ d)V(cz
+ d)fl
we applied
,
Hecke summation.
In the next section ($4: Analytic continuation of Eisenstein series) we deal with the question of Hecke summation. We shall show that in the case of a congruence subgroup l? (i.e., a discrete subgroup of SL(2,R)” which contains a principal congruence subgroup I’K[a] of some Hilbert modular group I’K. A deep Theorem by A. Selberg states that each discrete subgroup with a fundamental domain of finite volume and with at least one cusp is conjugate to a congruence subgroup.) Hecke summation always exists. We hence assume in the following that l? is a congruence subgroup. Let
be two vectors
of integers
b)
with
the properties
“j+/3j=2
Then
the Eisenstein Ea,B(~)
can be defined by He&e of the differential form
j=l,...,n.
fOT
series “ = ” c
N(cz
summation.
Cd=
+ d)YV(cz
We now
dza A fia
+ d)fl
return
to the symmetrization
A dz b.
Ya
We have w 1 A4 = w  N(cz
+ d)“N(cz
+ d)+
,
Chapter
152
III.
The
where
Cohomology
of the Hilbert
Modular
Group
1 ifjEa “j =
2
ifjEb,
1
ifjEa
0
ifjEb.
Pj = The results form
about Eisenstein
E(w)
“ = ”
c
series described
above show that the differential
w 1M “ = ” w . c
N(cz
+ dpv(cZ
+ d)+
MEIm\I’
can be defined by Hecke summation. 3.2 Proposition. the differential E(w)
“ =
”
Let w be one of the basis elements form c
c
wlM:=;li
1 N(cz
described
+ d) I”
in 3.1.
w 1M
Then
(s 7 0)
h4am\r
MEr,\l
exists.
We now have to investigate whether the differential form form E(w) is closed or not. A differential form
f(z)
dz, A dz,
y.
Adq,
(aubc
{l,...,n})
is obviously closed if and only if f( z ) is h o1omorphic in the variables coming from
b.
R.ecall:
In the next section ($4, Theorem 4.9) we prove the existence of a real number B such that adz>  VJY is holomorphic in all variables zj with /3j = 0. As a consequence, the differential form E(w) is closed if and only if the number B is 0. We also will obtain some information about the constant B: It is 0 if the number of all j such that
$3
Eisenstein
Cohomology
153
is less or equal n  2. In our situation
we have
aj = pj = 1
e
jEa
and m=#a+n=2n#b. This means that B is zero if #b22
or
m@ 1 N(cz
+ d) 129 )
$4 Analytic where
Continuation
159
of Eisenstein Series
a, /I E Z” are two vectors
of integers
with
the properties
Q#P,
4
b)
2r := “j
+ p.j
of j (f or our
is an even number, independent sufficient). The series converges
purpose
r =
1 would
be
the behaviour
at
for 2r+Res>2.
In this
section
exists and the cusps.
we want
we will
to prove
also obtain
that
precise
4.1 Remark. Let I? c I” group
of finite
index.
be a subgroup Then we have
E,‘:ak
c
3) =
in the case r = 1 the limit
information
which
about
contains
E,&>
SL(2,
s)lW
R)n
aa a aub
7
MEr\I”
where (f
the operator 1 M)(z,s)
. 1M = N(cz
is defined + d)“N(cZ+
by d)p
1 N(cz
This remark, which is of course trivial, allows of analytic continuation to the main congruence assume, unless otherwise stated, that
+ d) I+
f(Mz,s)
.
us to reduce the question group. From now on we
We determine a set of representatives of I’,\I’. A pair o) is the second row of a modular matrix (E SL(2,o)) unit ideal
(c, d) of integers iff it generates
(in the
cc, 4 = (1) * A necessary of course
condition
for being
the second
c=Omodq, But this condition may replace
(together
with
d=
row of a matrix
in l? = rK[qj
is
lmodq.
(c, d) = 1) is also sufficient,
because
we
Chapter
160
III.
The
Cohomology
of the Hilbert
Modular
Group
and hence assume b=Omodld
=+b=Omodq. From the relation
ad  bc = 1 we then obtain
a E 1 mod q.
Two matrices (:
;),
(;I
;)Er
define the same coset iff there exists a unit e E o*
)
elmodq
such that c’ = EC )
d’ = Ed.
In this case we call the pairs (c, d) and (c’, d’) associate have the explicit form of the Eisenstein series &,&,
3) =
c (c,d)=l,(c,dh
CEO mod
q,del
N(cz + d)TV@+ mod
1 N(cz
+ d) 12” .
q
The subscript (c, d)q indicates that the summation of representatives of associate pairs. This type of Eisenstein man.*
d)+
mod q. We now
series was considered
is taken only over a set
in 1928 already
by Klooster
(In this connection we should mention that the method of HeckeKloosterman could not be generalized to other modular groups, for example the important Siegel modular group. In a deep paper, Langlands developed a method which gives analytic continuation of all Eisenstein series on semisimple Lie groups. But it would be even more complicated to extract our special case from Langlands’ paper than to give the direct proof following Kloosterman.) Before we start with the analytic continuation we have to introduce more general Eisenstein series: Assume that (besides our level q) a further ideal a is given. We do not demand that a be integral. For two elements CO
* Kloosterman easily reduced by Maa%.
Ea,
do E a
actually merely considered the case /3 = 0. But to this case by means of certain simple differential
the general operators
case can be introduced
$4 Analytic
Continuation
we define
the Eisenstein
Ga,p(z;
S;(CO,
do);
161
of Eisenstein Series series (of level
q) as
a) = Gz,,(z; s; (co,~o); a) = I qcz + d)aN(cz + d)+ c
1 qcz
+ d) p
.
CECO mod qa mod qa,(c,d)q
d=do The
summation
is taken
(c, d) E K x K ,
over a set of representatives
(c, d) # (0,O) ,
with respect to the introduced associate if there exists a unit
out (for
that no condition z E H”) if 2r+2Res
and represents a + p = (2,.
an analytic .
.
)
E qa ,
CO
d
E qa
do
relation: Two pairs (c, d), (c’, d’) are called E E o*, E = 1 mod q, with
d’=Ed.
c’=m, We point converges
c
of pairs
of coprimeness
> 2 function
(2T
=
is demanded!
"j
+
on s (we
pj
E
This
series
22)
are interested
in the
case
2)).
4.2 Lemma. The group GZ+(2, K) = {A E GL(2, K) acts on the f(z,
s)
H
space generated by all Ga,p
Rvector
group
Because
by means
of the formula
(det A)2r+2S .f(Az,s)N(cz+d)“N(cz+d)+
PTOO~. It is not difficult to find an explicit Eisenstein series. For the sake of simplicity The
1 det A > 0)
G1+(2,
K) is generated
1N(cz+d)
I”
expression for the transformed we make use of the simple fact:
by the special
matrices
of the formula
one may even assume that a lies in a given ideal 4.2 is very easy. For example Ga,p(z
+ a; s; (CO,do); a) = GQ(z;
a. Now the proof
s;
(CO,
do +
coa);
a)
of Lemma
.
Chapter
162
G,,B(Z‘;
III.
The
Cohomology
S; (co, do); a) = N(z)“N(Z)~
Our next goal is to express of the G’s. For this purpose
of the
1N(z)
the Eisenstein
Hilbert
Modular
Group
12’ Ga,p(z; s; (do, co);a).n
series E as a linear combination
we need the notion of a “ray class
mod q”.
Notation: 1 = group of all ideals of K, ‘FI = group of all principal
ideals.
A (not necessarily integral) ideal a E Z is called coprime prime divisor of q occurs in the prime decomposition of a.
to q, if no
We denote by %I> c 1 the subgroup of all ideals which are coprime to q. We also have to define a certain subgroup R(q) of the group ‘H of principal ideals: A principal ideal belongs to X(q) i f an d only if it has a generator a! with the following two properties: a) cx > 0 (totally positive). b) The denominator of the ideal (CX l)q’
The usual proof of the finiteness that the group
is coprime
of the class number
to q, i.e.
h = #2/H
also shows
~tww is finite. Its elements
are the socalled
We also need the Mobius integral ideals. Let
ray classes
function
mod q.
p(a) which
is defined on the set of
a = p? . , . . . pz be the prime decomposition
of an integral
l44 =
WY o
ideal a. One defines if all Vi = 1 otherwise,
p(0) = 1. The Mobius
function
has the basic property
c44 PaCl={01
ifq=o ifq#o.
$4 Analytic
Continuation
163
of Eisenstein Series
After these preparations we can give an explicit expression of E as linear combination of the G’s. Introducing the Mobius function we may get rid of the condition of coprimeness in the definition of E, namely
c p(a)N(c*+d)w(c~+d)~ 1N(cz+d) 128 
~%,p(z,4 = c
The occuring ideals a are of course coprime with q (because d G 1 mod q). We obtain
a integral coprime
(c,d)+O,l) (c,d)E(O,O)
c
with
q
N(cz +d)V(cz +cl>+ 1N(cz +d)129 .
mod q mod a,(c,d)q
We now fix a ray class mod q
and consider the contribution
of this ray class to Eu,p(z, s):
Ea,p(d; z, 3) = C
h+
aEd
a integral
(c,d)=(O,l) (c,d)=(O,O)
c
qcz+qYv(cz+cl>+1N(cz +d)125 .
mod q mod a,(c,d)q
Of course we have
Ea,&,4 =
Ea,p(d;z,s>. c dMqL)/Wq)
Now we fix an integral ideal in our given ray class A
wEA,
a0Co.
Then every other ideal a E A is of the form a=yao,
y>O.
:
164
Chapter III.
&,,@; z,4 = c
Group
The Cohomology of the Hilbert Modular
p(a)N(yp+S)
aEA
a integral N(c’z
c (c’,d’)r(O,l) (c’,d’)E(O,O)
+
d)“N(c’z
+
cl’)+
1 N(c’z
+
d’)
I2e
.
mod q eo,(c’,d’)q
mod
The ideals q and aa being the property
coprime,
we can find a pair CO(= 0) , do with
= (0, 1) mod q (CO,do) G (0,O) mod a0 . (CO,
We now
obtain
&,p(d;
z, s) = N(ao)2(p+g)
do)
c
~(a)nT(a)2(‘+“).G,,B(z;
s; (CO,
and A C o* be a subgroup
of finite
do);
a~).
GA
a integral
4.3 Lemma. Let m C K be a lattice acting on m,
index
Axm+m (E, cZ)H Then if a runa
over a complete
system
EU. of representatives
of m
(0) mod A
the series c
I w4
I
,
0 > 1>
converges.
4.31 Corollary. The series
&A
a integral
defines an analytic finction
on the domain
Re s > 1.
Proof. We can choose the system of representatives such that (al,. . . , a,) is contained in a fundamental domain Q of A acting on Rn by (X,$+X&.
Such fundamental domains have been determined. The series can then be compared with the integral 1 vol(m>
J rEQ,IN(z)l21
( iv(x)
Ib dx1 . . . dx, .
This proof gives a little more than stated in 4.3, namely
cl
$4 Analytic
Continuation
4.32 Remark. The
of Eisenstein
(Notations
165
Series
as in 4.3)
limit
lilh((T  1) c’
1N(a) 1O
exists (and is unequal to zero). For our purposes we do not need the deeper result of Hecke that (s  1) c’
I N(a) r
has an analytic continuation as entire function into the whole splane. Analytic Continuation of the Eisensteiu Series G (as functions of a). We fist consider the simpler series fa,p(z; s; m) = C N(z + g)aN(Z gem
+ g)@ 1N(z + g) l2s
where m c K is any lattice, for example an ideal. The function fLy,p remains unchanged if we replace
z+z+a,
aEm,
and hence admits a Fourier expansion fa,p(z; s; m) = me(“‘/2)S(pa)
1
hg(y)e2”‘s(g”)
.
gEm*
Here m* denotes the dual lattice of m. The square root of the discriminant d(m) equals the volume of a fundamental parallelotope P of m. The Fourier integral gives the following expression for h,:
h,(y)= e(+9SkB) . = e(42)S(8) J = 1Ny ys
J
I
fcl,p(z, s; m)e2~is(gz)dx P
R” N(z)?IV(F)p
( N(z)
ls2’ e2niS(gz)dx
/iv(y”+q.
N(l  iz)m’N(l
+ ix)@
I N(l  ix) l28 e2rriS(gyr)dx.
R”
The integral splits into a product of n integrals of one variable. We first collect simple properties of this onevariable integral.
Chapter
166
III.
The
Cohomology
of the Hilbert
Modular
Group
4.4 Lemma. Put
such that
o+,B+Res>l. Then
the integral
m (1  it)W(l
h(y; (Y + s; p + s) :=
+ it)s
1 1  it I”
emitYdt
J cc3 converges (for arbitrary y E R). It has an analytic in fact morphic function into the whole splane, y # 0. Special values of h:
continuation as a meroas an entire function if
a)y=O: h(o;~+s;P+s) b)s=O(a,pEZ).
=

1)21("+p+28)
Onehasfory>O NY; a; P) =
where Pa&Y) for example
+ P+2s qa+s)qp+s)
273a
is a certain
h(Y;
polynomial
P; a> = eypa,p(y) in y which
Pcr,p(y>
= 0 if a 5 0
P@(y)
= (QT1)!Ya’
Basic estimate for h: Ifs a constant C such that
varies
in a compact
,
can be computed
explicitly,
if (y 2 1 . set of the splane,
there
exists
I h(y; a + s; ,f?+ s) 1~: CeIYI12 . Proof.
If we replace t by t,
we observe
h(y; a + s; P + s) = h(y; P + s; a + s) and hence assume y 2 0. For the computation of the integral at y = 0 and for the analytic continuation as well as for the basic estimate we may assume p = 0, because the integral only depends on a! + s and ,f3+ s. I Computation of the Integral at y = 0. Integration by parts gives h(0; a + s; s) = S/(CY 1). [h(O; a  2 + s + 1; s + 1)  h(0; QS 1 + s + 1; s)] ,
$4 Analytic
Continuation
of Eisenstein Series
167
if 01 # 1. The same recursion formula is satisfied by the rexpression in 4.4. It is therefore sufficient to treat the cases (Y = 0 and a! = 1. In both cases the transformation t2 + 1 = 5l reduces
the integral
II Analytic
to an ordinary
Bintegral.
Continuation.
The analytic continuation will follow from integration. We hence define the integrand (1  ity(l+
a deformation
t2)se“ty
)
t , Im t < 0. (It looks the lower halfplane, beThe only problem is the
= esh3(l+ta)
.
We define log(1
t2)
+
= log 1 1+
t2 1 +i
arg(1
+
t2)
where arg(l+P)
:= arg(t
7r/2 37r/2 This
definition
1)
arg(1
2)
q(l
t2) = + t2> is
Let
t
3x/2,
< arg(ti)L
7r/2.
be a point
three
 i),
properties:
0 if t E R . continuous on the domain
{tEC 3)
arg(t
<arg(t+i)
. We now decompose
our integral
into two parts:
a) The integral along the circle around i is of course an entire function of s. b) We compute the jump of the integrsnd at the critical line if we pass it from the right to the left halfplane: The jump of the function (1 + t2)s at a point
t on
(it ~]l,oo)) is  eaia] = 2isin7ris.
the critical
axis
(1+ 1t 7)s[p5 The contribution
of the two vertical 2i sin 7ris .
J
] t 12)’ .
to our integral hence is
lines
(1  it)a(l+
(l+
] t 12)seitydt,
where the path of integration is the vertical line on the right hand side (starting from a point it0 , to > 0). This integral again defines an entire function of s. III The basic estimate is an immediate defines the analytic continuation.
consequence
of the formula
IV The special value of h at s = 0: F’rom the residue theorem positive y h(y; a!; p) = 2 ri
tF&s(l  it)“(1
The residue is zero if a 5 0. If cr 2 1 it is (i)” a,1 in the expansion (1 + it)beitY
= 2
a,(t
+ it)beity
which
we obtain for
.
times the Taylor coefficient
+ i)” .
v=o All these Taylor coefficients are obviously products of ev with certain polynomials in y. Their trivial computation completes the proof of 4.4. Cl As a consequence of Lemma 4.4 we obtain the analytic continuation of the series fa,a :
$4 Analytic
Continuation
169
of Eisenstein Series
4.5 Proposition. Let m c K be a lattice in K. The series
fa,j3(z; s; m) = C N(2 + g)aN(T
+ g)+
I N(z + g) I”
has the Fourier expansion
. c hg(y)e2”“s(g”) , gem*
vol(p)e(“‘/2)S(fla)
where h,(y) =I Ny /12s2r ~hg(2?rgjyj;oj+s,~j+S). j=l
This Fourier series defines an analytic continuation of fa,~(z,s; m) as meTomoTphic function into the whole splane. The only poles come from the zero Fourier coeficient, i.e. fa,p(z; 9; m)  vo1(P)e(“‘12)S(B“). r(2T
) Ny j12s2r (27r)” +
r(aj
29
+
1)
* 212(‘+a)
s)r(@j
+
9)
1
is an en&e function of 9. (Recall:
2T
I= ‘Yj + @j E 22)
We now express the Eisenstein series G+(z;
s; (CO,
do);
a) :=
I
N(cz + d)a.N(c~ + cl)@ 1N(cz + d) 12s
c csamodqa d=dr, mod qa,(c,d)q
by means of the function fa,a. The contribution of all pairs (c, d) with c = 0 is zero if CO# qa and I c d=do
iv(d)2’
1N(d) 129
modqa dq
if co E qa. The summation is taken over a set of representatives of all d=domodqa,
dfo,
with respect to the “associate relation”: Two elements d, d’ are called associate mod q if there is a unit E, E E 1 mod q, with d’ = cd. If we introduce the number 1 if COE qa 6 = 6( CO,qa) = 0 elsewhere,
170
Chapter
III.
The
Cohomology
of the Hilbert
Modular
Group
we obtain
Ga,p(z; s; (CO, do); a) =S 
c dsdo
+
1 N(d)
mod 4
C' CEC,,
fe,j3(cz mod
12(p+g)
qa
+
d0;s;qa).
qa
cq
We now replace the fa,a by their Fourier expression computed in 4.5. The volume of a fundamental parallelotope of m = qa is
where dK denotes the discriminant of K. From 4.5 we obtain c’ CECO
f&cz
mod cq
+ do, s; qa)
= n/(qa)drce(“‘/2)S(P“)Ny’252’.
qa
c=co c”g”d qa gc(qa)*
If we collect in G,,p all t erms with fixed cg we obtain the Fourier expansion Ga,p(z;q(co,do);
a)
=
c
~~(y,s)e~~~~(~+)
,
gl*
where the Fourier coefficients are given by the following formulae a) g = 0: n q2r + 2s  1) * 212(‘+8) s> = 6. n F(“j + S>ryPj + s) j=l
UO(Y,
.
c’ CECO
1 NC mod
ll282r
qa
cq .
C’ dsd,
1 mod
N(d)
I2(r+s)
+(2~)“N(qa)dKe(“i/2)S(B‘Y)N?/1292’.
qa
ds
b) g # 0: U,(y, S) = N(qa)dKe(““/2)S(B(r)
c g=ed,dE(qa)* czco mod
. e2(d.do)
. N
129277
Y
1NC 112s2r qa,cq
. fi j=l
h(2ngjYj;
“j
+
3;
pj
+
S)
.
$4 Analytic
Continuation
171
of Eisenstein Series
The sum is a finite one which can be estimated by a constant times a suitable power of 1 Ny I. So the interchange of the summation is justified. We now obtain: 4.6 Proposition.
The difference
of the Eisenstein
series
and its zero Fourier
coeficient G&Z; has an analytic
s;
4  UO(Y,
(co,~o);
s)
continuation as entire function of s into the whole
splane.
Remark. If one makes use of the fact (which we did not prove) that the series c’
I N(d) r
mod qa dq
d=d,,
admits an analytic continuation as meromorphic function into the whole splane, we obtain: The series Ga,p, Ea,p admit analytic continuations into the whole splane as meromorphic functions. We now assume and
Z!T=(Yj+pj=Z
CK#P.
We want to investigate the Eisenstein series Gm,b if we approach the border of absolute convergence s = 0. Because we assume (Y # p, T = 1, we have 5 0 or
Ctj
/3j
5 0
for at least one j. This implies that 1 r(‘yj
+ S)
1 Or
r(Pj
+
S)
has a zero at s = 0. On the other hand the limit
exists (4.32). From our explicit formula for the zero Fourier coefficient we now obtain
Fyo ~o(Y, 3) = A + B/Ny , where A and B are constants. We collect the properties of the constants A and B which were needed in §3.
Chapter III. The Cohomology
172
4.7 Lemma. We
of the Hilbert
Modular
Group
have
lim ao(y, s) = A + B/Ny S+0
with certain Teal numbeTs A, B. The constant j with Q!j = pi = 1 is less OT equal n  2. From
the Fourier
expansion
B is .zeToif the numbeT of all
from the results 4.4 about
of Ga,p and especially
the function h(y; ol; /3) we now obtain: 4.8 Theorem. Let (Y, ,d be two vectors
a # /I
and
of integers
ckj + @j = 2
such that
(1 5 j 5 n) .
The limit lim G,,p(z;
3;
S+0
exists
(CO,
a)
do);
and has a Fourier expansion of the following A + B/Ny
+
type:
c a,P,,B(gy)e2”S(lgl’)e2”‘S(gZ) Sea* ,s#O
with 191 :=
IsnO*
(IslL~~~
The coeficienta a, E C can be estimated by the suitable power of [Ngl. The functions Y
are certain
WY)
of a constant
pTOdUCt
and a
. %B(Y)
polynomials.
We do not need the explicit form of the coefficient function P,,p(y). We only notice that the calculation of the special values of h(y; (Y; 0) = h(y; 0; cx) in 4.4 shows 4.81 Remark. Assume @j = 0 for some j. the variable yj and moreover pa,&/)
4.82
Corollary. Assume lim
O+0
=
0
if
Yj
Then
y
b)a#PThen
the limit
E,,jj(z) := liio c N(cz+ a)QN(cZ + a)@IN(cz + a)l“” r,\r exists. If M
E GL(2,
K)
is a matrix with
totally
positive
determinant, the
function (E,,pIM)(z)
= N(cz
+ a)“N(E
+ a>+
has a Fourier expansion of the following (E,,pIM)(z)
= A + B/Ny
 E,,p(Mz)
type
+ c
a,P(gy)e2”s(lglm’)e2niS(g2)
,
gEtQ where
A, B denote
real numbers, the function
Y
(NY)  P(Y)
is a polynomial and the numbers ag have moderate growth, i.e. they can be by the pTOdUCt of a constant and a suitable power of INgl. The constant B is zero if the number of all j with aj = @j = 1 is less OT equal
estimated n  2.
The number
A is I if M
is the unit
matrix have:
is not equivalent to 00. we fuTtheTmoTe Assume
/3j = 0 for
some j.
Then
the function
J%,&) is holomorphic
but zero if the cusp M‘(00)
B/NY
in zj.
We only have to put together what we did in this section: We expressed &,B as a sum of Ga,p (with real coefficients). We proved that the group GL(2, K) act s on the space which is generated by the Ga,p over R (4.2). Up to the statement about the constant A, Theorem 4.9 is hence reduced to the Ga,p. This last statement follows from the formula Proof.
174
Chapter
III.
which is easily verified Fourier expansion. The
for the limit
The
G,,p
Cohomology
instead
of the Hilbert
of &,b]M
by means
can be computed in the same way as in the case of holomorphic series of weight 2r > 2 (Chap. I, 5).
§5 Square
Integrable
Modular
Group
of the
Eisenstein
Cohomology
The results of $3 (including $4) will allow us to write each cohomology class of H”‘(I) as the sum of an Eisenstein cohomology class and the class of a square integrable differential form. The latter classes can always be represented by square integrable harmonic ones. The theory of square integrable harmonic forms runs similar to the case of a compact quotient. The method developed there ($1) will give the complete determination of Hm(I’).
We denote
by
the subspace of all cohomology classes [w’] which square integrable (closed) differential form w, i.e. w = w’ + o!d The form Of course
w” needs not to be square “square integrable” refers
=
(
integrable. to the Poincark 0 *. .
The
aim of this
5.1 Proposition.
section
is the proof
metric
) .
2
0
by a
.
YT2
h(z)
can be represented
Yfl
of the following
two propositions.
Let Kl,..*,Kh
be a set of representatives
of the cusp classes. The Eisenstein cohomology
dejined in 13, maps isomorphically onto the image under the natural restriction map H” F)

&H”(L.,). j=l
$5 Square
Integrable
Up to now
Cohomology
175
5.1 has been proved
in the cases n 5 m 5 2n  2.
5.2 Proposition. In the case m > 0 we have
Remark: In the case m = 0 Proposition
5.2 is not true,
one has
Ho(r) = l&(r) = H&(r) E C. (We have H&(P) = HO(P) by d e fi ni t ion and this definition is necessary if one wants to have 5.1. On the other hand the constant form w = 1 is square integrable because Hn/l? has finite volume with respect to the invariant measure w A* w. This implies H’(P) = H&,(P)). The proof of the two propositions depends on a good knowledge of the square integrable cohomology. The latter can be investigated by means of two important general theorems about complete Riemannian manifolds (which we explain in App. III without proofs).
A) Each square integrable cohomology square integrable differential form. B) Each square integrable harmonic We denote
class can be represented form
by a harmonic
is closed.
by %&l(r)
the space of all square The
two theorems
integrable
harmonic
A and B above
forms
of degree
give a surjective
m.
map
but in contrast to the cocompact case this map need not to be injective! The space Z,“,,(P) can be determined (because of B) in precisely the same way as in the cocompact case. We only have to check which of the harmonic forms occurring in $1 are square integrable. 5.3 Lemma. a) The universal are square integrable.
cohomology
classes (generated
by dzi A &i/y:)
b) Let f~ be a (holomorphic)
ailbert
w2,...,2)i
modular w = f(2)d.q
is square
integrable
form.
The
differential
A . . . A dz,
if and only if f is a cusp form.
form
Chapter
176
III.
The
Cohomology
PTOO~. a) Up to a sign w,, A *wa is the invariant has finite volume.
of the
Hilbert
Modular
volume element,
Group
but Hn/l?
b) One has w A *W =I f(z)
I2 Euclidean
volume element ,
=, JWA*53
hence
where the brackets on the right denote the Petersson scalar product, introduced in Chap. II, 51. W e h ave shown that this converges if and only if f is a cusp form. 0 5.4 Theorem.
(Compare
1.6) Let I’ c SL(2, R)”
that the extended quotient of H”/I’ is compact compact. We have a “Hodge decomposition”
where
be a discrete but such
that
subgroup
H”/I’
such
is not
1) in the case p + q # n c
7gg(r) = 7ig$ =
0
i
(wa =
ifp=qln
cwa
#a=p
dzcz, A Gz, y2 01
/\
elsewhere
. . . A dza$
%
),
QP
2) in the case p + q = n
7f;:(r) E a:;,
@ bC{l
(Recall: [I’,
~10
~r~,(2,...,2)io.
,...,nl #b=q
denotes the space of cusp forms.)
As a consequence of 5.4 we obtain 5.5 Lemma. Let w be a square integrable harmonic form of degree m > 0. If 00 is a cusp of l?, there exists a I’,invariant form CYsuch that w=da!.
$5 Square
5.51
Integrable
Corollary.
177
Cohomology
Assume
of the two mappings
m > 0. The composite h
‘FI,“,,(r)
+
Hm(r)
+
$
Hm(rlCj)
j=l
is zero. 5.52
Corollary.
H&(r)
n H;,(r)
= (0)
if m > 0.
PTOO~. We show that a square integrable harmonic form defines the zero
class in H”(I’,),
where roe is the stabilizer of the cusp co.
1) Universal classes: The forms
axe r ,invariant
(but not I?invariant). One has
d(ai) = dxiy: dyi = :Wi p t hence 4%
A w,, A . . . A warn) = (1/2i)waI A.. . Aw,,
ifm>l. 2) Classes coming from cusp forms: We may restrict ourselves to the case f(z)dzl
A . . . A dz,
,
where f(z)
= C
age2TiS(gr)
is a cusp form of weight (2,. . . ,2). We have a,#0 We integrate f(z)
*
g>o.
with respect to the first variable g(Z) = C
ag/(27rigl)e2”“s(gZ)
The form g(z)dz2
A . . . A dz,
.
Chapter III.
178 is I’,invariant
The Cohomology of the Hilbert
Group
and one has d(g(z)dz2
A..
The proof of 5.5 actually property: We consider a sequence
. A dz,)
= f(Z)dZl
gives a little
more,
of Coofunctions
A..
. A dz,
namely
0
.
a certain
approximation
on the real line
‘Pk : R +[O,l], with
Modular
k=1,2
)...
the property 1 cPk@) =
1 0
iftk+l
and I cps>
IS 2 *
We define
+k : H* 
P, 11
by 4k(z>
In the notation
of Lemma
=
5.5 we now
+‘k(Ny).
consider
and ‘dk := We certainly
have
(pointwise convergence). Lemma 5.5 gives a little
5.5~ Remark. proximation
d((Yk).
With result:
O!k 
Q!
Wk 
‘d
But the explicit more, namely
the notations
construction
of Lemma
during
5.5 we have
the proof
the following
(UC = {z E H” 1 Ny > C} , C > 0)) where
,6 is any square
integrable
harmonic
form
of complementary
degTee.
of
ap
55 Square Integrable
179
Cohomology
We leave the proof
to the reader.
For the proof of Propositions the Poincar15 duality.
5.1 and 5.2 we need a further
tool,
namely
which Recall (see App. III): The de Rh am complex has a certain subcomplex consists of all differential forms with compact support. The cohomology groups of this subcomplex are the cohomology groups with compact support which we denote by
One has a natural
linear
which
neither
is in general
mapping
injective
nor surjective.
H,O(I’) The
following
two theorems
duality):
1) ( Poancare’
Obviously
= 0.
are explained
(but
not proved)
in App.
III.
The mapping
(w.4  JH”/rWAw’, w, w’ are closed differential forms, the first one with induces a nondegenerate pairing where
H,“(r) We especially
x Hz”“(r)
c.
have dim Hr(l?)
2) TheTe

compact support,
exists
a linear
=
dim H2”“(I’)
.
mapping

6: &H”(L+,)
H,“+‘(r)
j=l
such that the long
sequence h
. ..
is exact.
f
H,“(r)
+
Hyr)

@ H”(l?Kj) j+l

H,“+‘(r)

...
Chapter III.
180
The Cohomology of the Hilbert Modular
Group
We use this sequence in the case m = 2n  1 and obtain the exact sequence Pnl(r)

0
Pl(rnj)
4
Hp(r)
+
P(r).
j=l
From
~,2yr) E Ho(r) E c H2yr) 2 H;(r) = 0
we obtain: The image of
H2nl(r) +6 H2nl(ry) j=l
is a subspaceof codimension 1. This completes the proof of Proposition 5.1 in the case m = 2n  1. The cases m 2 n now can be treated by duality: From the surjectivity
of the restriction map Hyr)

$Hm(rKj) i
in the case n 5 m 5 2n  2 and from the long exact sequence we obtain that
,m+l(r) q P+l(r) c
is injective in those cases. Dualizing this result we obtain: 5.54 Remark. The map
H;(r) + P(r) is surjective if 1 5 m < 72. The image of this mapping is of course contained in the square integrable cohomology. We obtain
wyr) = H;,(r)
if 1 5 m < 12.
From Lemma 5.5 we finally obtain that
Hm(r) 
&H”(rKj j j=l
is the zero mapping if 0 c m 5 n.
$5
Square
Integrable
This jusitifies
181
Cohomology
the definition
H&(P)
= 0 in these cases!
The proof of Proposition 5.1 is now complete. From 5.51 and from the long exact sequence we conclude furthermore that the square integrable cohomology is contained in the image of the cohomology with compact support if m > 1. Hence both are equal and the square integral cohomology is precisely the kernel of the restriction map (5.1) ( if m > 1). Now Propositon 5.2 follows from 5.1. Our next goal is to determine the kernel of the mapping
5.6 Lemma.
a) is injective
The natural
if m < 2n
b) is the zero mapping This
mapping
if m = 2n.
means
7i,m,,(r) if m am,
PTOOf.
=
{ 0
ifm
< 2n, = 2n.
Let
be a square integrable harmonic form whose cohomology class in Hm(I’) is zero. Prom the existence of the Poincare pairing it follows
J wAa=O, where (Yis a compactly supported closed differential form of degree 2n  m. We want to show that in the case m < 2n this implies w = 0, or equivalently
J wA*z=o. The convergence of this integral follows from the explicit description of the square integrable harmonic forms. The idea now is to approximate *sj by compactly supported closed forms. We now apply Lemma 5.5 to *G instead of w. We may apply this lemma to write *W as the derivative of a certain form in a small neighbourhood of an arbitrary cusp class. These differential forms can be glued together to one form QLby means of “partition of unity”. The result of this construction is a form p whith compact support such that
Chapter
182
III.
The
Cohomology
ij  /3 = dcu.By means of the approximation construction: There exists that
a) %pk
a sequence of compactly
of the
Hilbert
Modular
Group
lemma 5.53 we may refine this
supported
differential
forms
@k such
= dak
The integrals as desired.
in the sequence vanish by assumption.
We obtain
w = 0 cl
Final Remark: We now have the complete picture of the cohomology and also of cohomology with compact support (by means of Poincare duality) and the square integrable cohomology. There is also the notation of the cuspidal cohomology. Let lattice
f : H” + C be a continuous function which is periodic with t C R”. We call f a cusp form at 00, if the zero Fourier coefficient
R”,t
f(z)&
respect
to some
. ..&I
J vanishes
(this
coefficient
A Iinvariant
is a function
differential
form
of y). w on Hn is called M E SL(2,
UIM, are cusp The
forms
a cusp K)
form,
if all the components
of
,
at 00.
cuspidal
part HcmUsp (r)
consists of all cohomology shown that each cusp form
The explicit
universal description
classes which may is square integrable,
forms w. , Q c { 1, . . . , n  1) , are obviously 5.4 we obtain
fc:,,(r) if m = n and
be represented by a cusp hence we have
=
not cusp
form.
forms.
It can be
Prom
the
$ P, (2, . . . , a0 bc{l,...,n)
0 elsewhere.
36 The Cohomology
of Hilbert’s
Modular
Groups
We only have to collect the results of the previous sections to get a complete of the cohomology of the Hilbert modular group, more generally of congruence
picture groups.
$6 The
Cohomology
of Hilbert’s
Modular
183
Groups
The formula in the Betti and Hodge numbers involve several invariants of those groups like volume of a fundamental domain, number of elliptic fixed points of given type and certain Lseries coming from the cusps. All these invariants can be computed in case of real quadratic fields.
In the following, l? denotes a congruence group, and ~1, . . . , ICY representatives of the cusp classes. In the Sects. 3,4,5 we investigated the restriction map. The most difficult part of the theory was the construction of an injective homomorphism
&H”(r.j) t
space of I’invariant
differential
forms
of degree
m
j=l
in the cases n 5 m < 2n. The
image
of this map
is the space of Eisenstein
series W,
m> 
As the Eisenstein series did not converge absolutely, we had to do the tedious job of analytic continuation ($4). Not all the Eisenstein series are closed differential forms. The subspace of closed forms has been denoted by E0(r,
m>
c
E(r,
m)
.
In case m = 2n  1 this subspace has codimension l.In the cases n 5 m 5 2n  2 both spaces agree.The natural map of &(I’, m) into the cohomology group of I’ is injective and hence defines an isomorphism to a certain subspace
Hg(r) c fvyr) . The case m < n could been forced to define
be treated
Hzm
= 0,
by means
if
of Poincar& duaIity.We
have
O<md z + z’Go where Gz,, := (G;,rC)t . 4) Toeachr= (or,..., w,) E C(r) there is associated a connected submanifold F, c Xc of codimension r. In coordinates we have F, n (C”), = {z~zr=...=zr=O},if0=(wr ,..., w,,w,+r ,..., wn). 5, D ‘= Udimrcl F, is a divisor with normal crossings in Xc. For r = wr) E Cc’) we have: ( Wl,...,
F, = F,, n . . . n F,, But if (wr , . . . , 21~) # C(‘),
then
:
.
F,, n . . . n F,, = 0.
$7 The Hodge Numbers of Hilbert Modular Varieties
201
6) Let I’ c I?K be a congruence subgroup of some Hilbert modular group r~; let t be the translation lattice and A the group of multipliers of r. Then there exists a simplicial complex C having the following properties: (i) ICI := u (T = (R”,) u (0) oEC (ii) A acts on C, i.e. g E C, .5 E A j c0 E C (iii) 0 E C, E E A, E # 1 * dim(a n ~7) 5 1 When replacing r by a suitable subgroup I” c r of finite index, we can attain dim(o n ~0) = 0 in (iii). 7) There is an open neighbourhood U(D) c Xc of the divisor D and a biholomorphic mapping R:
?r : where
U(D)\13
N l&/t
UC := {z E Hn 1Ny > C > 0)
8) A acts discontinously and freely on U(D), such that: (i) x E D, e E A =+ EZ E D, i.e. A acts on D. (ii) n : U(D)\D II UC/t is equivariant with respect to the natural action of A on UC/A. 9) Consequently U(D)/A is a complex manifold, Y, := D/A a divisor with normal crossings in U(D)/A. We have:
(U(D)\D)/A=(U(D)/A)\Y, is an open neighbourhood
=
of the cusp {co}.
f:U(D)/A+Uc/Lu{oo}
uc/roo cH”/F We define a mapping
cXr
by f(Ym) := {oo} and by defining fj(U(D)/A)\Y, as the induced isomorphism r* : (U(D)/A)\Y, 11 Uc/I’oo. Then f yields a resolution of the singularity {oo} of Xr with the properties described in the introduction. 10) The action of A on U(D) is such that each E E A gives a rise to a biholomorphic mapping
If I” E I’ is chosen as in 6. with arlaa = (0) for E # 1, then F, rl Fe, = 0 for E # 1. To see this let 0 = (~1, . . . , r~,.) and .zcr = (vi, . . . , u:) and 7 = (WI ,...) wr,w: )... , wk); then r can not be a simplex of Ct2’), since that would contradict r n ET = (0). Thus:
F, n F,, = F,, n . . . n Fur n Fv; n . . . n Fv; = 0 Therefore
we have FJA
z F,
202
Chapter
III.
The
Cohomology
of the Hilbert
Modular
Group
for each u E C. Especially Fvi /A II Fvi for each lsimplex v;. Consequently Y, = D/A is the union of smooth divisors DI, . . . , D,; moreover arbitrary intersections of these divisors are connected, since for suitable vik we have: Vii, *  , Dil n . . . rl Dik N Fvi, rl. . . rl Fui, II F, if r = (vir,...
,v&)
E d”)
and Dil
n.. . nDik = 8
if r $ Cck). Obviously we have a 1:l correspondence between the lsimplices vr , . . . , vr of Z/A and the smooth divisors D1, . . . , D,. The simplices (uir,. . . ,uik)R+ correspond to the submanifolds of codimension k given by Dil n . . . fl Djk, which coincide with the connected components Y& of Yk (see above).
Appendices
I. Algebraic We
Numbers
give a brief introduction,
proofs, to the theory of algebraic numbers.
number is called algebraic if it is a root with rational coefficients
A complex polynomial
c&an+... We may
without
+ma+ao
assume
that
ajEQforO<jIn,
=o,
the polynomial
polynomial
with
rational
P E &[z] is called
Gz#O.
is monk, i.e. a, = 1.
If a is a root of a manic polynomial integers we call a an (algebraic) integer. A manic
of a nonvanishing
whose
coefficients )
P(u)
coefficients of minimal
are rational degree
=o
minimal polynomial of a.
ALL The minimal polynomial of an algebraic number is uniquely determined. It is separable, i.e. it has no multiple root. The minimal polynomial of an algebraic integer has (rational) integral coeficienta.
Notation.
The degree of an algebraic number is the degree of the minipolynomial. The different roots of the minimal polynomial are called conjugates of an algebraic number a. We often denote them by
mal
&)
T*Y acn)
(u is one of them)
.
AI.2. The set a of all algebraic numbers is a subfield of C which contains Q. The set of all algebraic integers z is a aubring of a with the property
TnQ=Z.
Appendices
204
AI.3. Let P be a manic polynomial with algebraic coeficients (P E QI[x]). The roots of P are algebraic. If the coeficients are integral (P E z[x]), the Toots are algebraic integers. Number Fields. Let K be a subfield of the field of complex numbers (Q C K c C). We may consider K as a vector space over &. K is called an algebraic number field if the dimension of this vector space is finite. This dimension is called the degree of K and denoted by n = [K:Q]
= dimeK.
(One can always define the degree of an arbitrary field K with respect to a subfield k. [K : k] = d lrnk K 5 c~.) The elements of algebraic number fields are always algebraic numbers and each algebraic number is contained in some algebraic number field K. The smallest K which contains a is denoted by K = Q(a). Conjugate Fields. Let K be an algebraic number field of degree n. An imbedding of K into the field of complex numbers is a mapping cp:K with
t
C
the properties
da) = a
1)
for a E & ,
p(a i b) = y+) i p(b). The image of K is a subfield of C isomorphic
to K, K
N b p(K).
AI.4. An algebraic number field K of degree n admits precisely imbeddings into the field of complex numbers. We usually
arrange
the imbeddings K + a&),
K(j)
n different
in a certain order and denote them by c C , j=l,...,
n.
AI.5. If a is an element of K, the images under the imbeddings are conjugates of a. Each conjugate occurs under these images with multiplicity [K : Q(u)] = [K : &]/degree(a).
I. Algebraic
205
Numbers
(We have degree(a)
= [Q(a) : Q].)
Notation.
the n different
Consider
imbeddings
KK(j)
,
Sa = Sqf$a)
l<j
b)
0 = Zal + . . . + Za,
The discrimiiant
.
of K is by definition the discriminant of the lattice o, i.e. dK := d(o)
(= d(al,.
. . ,a,)>.
(The discriminant is a very important invariant of K. It can be shown that always d E 0 or 1 mod 4 and that for given d there exist only finitely many number fields with discriminant d, especially: n f co + d t 00.) Units. The invertible elements of o are the socalled units of K o* = {E E 0 1 E # 0 ) .5l E 0).
Special units of K are the roots of unity which are contained in K W={<EK
Of course o* is a multiplicative
1 t+,es=
The constructed
(gik). w~,w:
(wi,w:
>)l MnP(D)
such that wg
= WAW’
for all w E IMP(D)
,
w’ E APp(D).
The star operator is invariant with respect to orientation preserving motions 9 *(cp*w)
=
cp*(*w).
One has *(*w)
= (1)PnSPW.
The codifferentiation 6: ML(D)
+
(= 0 if p = 0)
Mzl(D)
is defined by 6 := (l)np+n+l
* d* .
The LaplaceBeltrami operator is A : M&(D)

MS(D),
A := d6+6d. Of course 6 and A commute with orientation preserving motions. (More generally: If y3 : D’ + D is an orientation preserving diffeomorphism we have p* o As = Apes .) XI. Hermitean Metric. A Hermitean metric is a complex matrix h with the property X’ = h. Each n x nHermitean matrix defines a real 272x 2n symmetric matrix g which is characterized by Z’h.z
=
a’ga,
where a’= (q,y1,...
7ZTl,Y7l
1.
231
III. Alternating Differential Forms
We say that g comes from the Hermitean matrix h. A Hermitean metric on an open domain D c C” is an n x nmatrix h of Cmfunctions on D, such that h(z) is Hermitean and positive definite for all z E L). The associate 2n x Bnmatrix g is a Riemannian metric. We have det h(z) = +dm. The fundamental form may be written as wg = mdq
Adyl A...Adz,Ady,
= &(det
h)dzl A &I A . . . A dz, A d!Z, .
Let now cp : DA
D, D’ c Cn open,
D’ ,
be a biholomorphic mapping, h a Hermitean metric on D’ and g the associate Riemannian metric. Then the pulled back Riemannian metric ‘p*g on D comes from a Hermitean metric, namely from
Here &((p,z)
denotes the complex Jacobian.
The LaplaceBeltrami
operator A
= d*d*
+ *d*d
(the real dimension of C” is even) usually does not preserve the decomposition Mm(D) = c Mpsq(D).
p+q=m Therefore one also considers the operators
•=a*8*+*8*a  ii =a*a*+*a*a which map MPfJ(D) into itself. This follows immediately from the following fact: In the case of a Hermitean metric the star operator maps (p, q)forms to (n  q, n  p)forms .+ : MM(D)
+
jtrfnq+p(D) .
XII. Kiihlerian Metric. A Hermitean metric h on an open set D c C” is called locally Euclidean at a point a E D if h(u) is the unit matrix and if the first partial derivatives of h vanish at a.
Appendices
232
Definition. A Hermitean metric h is called Kihlerian if it is locally equivalent with a Euclidean one, i.e.: For each point a E D there exists a biholomorphic mapping of an open subset U c C” onto an open neighbourhood V of a in D such that the pulled back metric cp*h is locally Euclidean at b = p‘(a). For any Hermitean
This differential mations cp, i.e.
metric
form
h we may consider
is invariant
with
respect
=
cp*Q(h).
R(cp*h)
Remark. In the case of a KZhlerian
metric dS2 =
The
converse
is also true
the (l,l)form
to biholomorphic
transfor
h we have
0.
but we do not need this.
Proposition. In the case of a KihleTian
metric
we have the identities
A=20=2Ii.
Corollary.
The LaplaceBeltrami M;(D)
We make the proof:
operator
A preserves
=
Mzp(D).
c p+q=m
use of this proposition in Chap. Let us consider the operator L
: MP,P
+
III,
the double
$1 and therefore
graduation
we indicate
MP+l,P+l
L(w)=RAw, where sition
R denotes the K&ler form (see above). are a formal consequence of the relations Lo*a* Lo*a*


*a*oL *a*oL
The
=
ia
=
8.
identities
in the propo
We leave this reduction to the reader. The advantage of the latter relations is that they involve only first order derivatives. From the definition of the
III. Alternating
Differential
233
Forms
K&ler property it follows that such a relation, which is invariant under biholomorphic transformations, has to be proved only in the case of the Euclidean metric h = E = unit matrix. In this case the relations can be verified by direct calculation. Example. Each Hermitean metric h on a (complex) ldimensional domain D C C is Kihlerian.
The de Rham Complex XIII. Differential Forms on Manifolds. Let X be a topological space. We always assume that X is a Hausdorff space with countable basis of its topology. A differentiable structure on X is a family Qj
ZUj +
Vj 3 Uj C X open, Vj C Rn open,
of topological mappings with the properties
a>
x+4, Qj O(Pi1 : Qi(Ui
is a P’diffeomorphism
fI Uj)
+
Qj(Ui
n
Uj)
for all (i,j).
The space X together with a distinguished differentiable structure is called a differentiable manifold of dimension n. A differential form of degree p on X is a family w
=
(w)
Wi
7
E M’(Vi)
3
such that the formula (Qj
0 Qi’)*Wj
=
Wi
holds on Qi(Ui fl Uj). We denote by i@(X) the space of all pforms on X and by M&(X) the space of all CODpforms (i.e. all wi are Cm). We may identify a function f : X + 4: with the zero form (fi)
3
.fi
=
f
1 uiOQ:l
*
The function f is called Cmdifferentiable if all the fi’s differentiable. We hence may identify M&,(X) and C”(X)
= {f:X+C
1 fist”}.
are C”
Appendices
234
There are natural
mappings MP(X)
x MQ(X)
4
Mp+Q(x)
(w,w’)l4wAw’, (W
A W’)i
:=
Wi
A W:
and M&(X)A
itdgyx>
(Ckd)i
:=
CJTWi
)
e
The sequence ... is the socalled
M&(X)
d > AIgyx>
de Rham complex
The de Rham cohomology C) are defined as HP(X)
(complex
groups

...
means: d o
d
(they are actually
=
0).
vector
spaces over
:= cyx)/Byx)
where
By the theorem
G’(X)
= ker(M,&(X)
BP(X)
= im(Mcl(X)&
of de F&am
there
exists
d b iL@l(X))
a natural
HP(X)“+ where
W(X,
C) denotes
the singular
M,&(X)). isomorphism
HP(X,
cohomology
C)
groups
with
coefficients
XIV. Real Hodge Theory. The Hodge theory is a powerful the de Rham cohomology groups in the case of a compact A Riemannian
metric
g on the differentiable
g = (gi) , such that the transformation
gi Riemannian
tool to compute manifold: X is a family
metric on Vi
formula (Cpj
is valid on pi(Ui
manifold
in C.
O Pil)*gj
n Uj). If a Ri emannian * : MP(X)
f
(*W)i
:=
=
Si
metric is given, the star operator APP(X) *(Wi)
III.
Alternating
is well
Differential
defined.
235
Forms
We therefore
may define
A : ML(X)
+
(Au)~ The
kernel
One of the main
results
=
operator
M,&(X)
= Awi.
of A is the space of harmonic tip(X)
the LaplaceBeltrami
forms.
ker(M,&(X)
A b M&(X)).
of the real Hodge
theory
states:
Assume that X is compact. Then each harmonic form is closed. The natural mapping ‘HP(X) HP(X) is an isomorphism. Notice: If w is harmonic
then
*w is also harmonic.
We obtain
(for a compact
manifold!) w harmonic
XV. Integration
M
&J = 0 and d(*w)
= 0 .
of nforms.
An nform
w = fdxl
A . . . A dx,
on an open domain D c Rn is called integrable with respect to the Euclidean measure:
if the function
f is integrable
Notation. J,w
:= kf(x)dx+.dx,.
If
cp:D’ is an orientation
We hence
preserving
may generalize
+
diffeomorphism,
the notion
D we have
of an integrable
nform
w and the value
to an arbitrary oriented differentiable manifold. Here “oriented” means that all transition functions 'pj 0 (pi1 are orientation preserving. A differential form w of arbitrary degree p on an oriented Biemannian manifold (X, g) is called square integrable, if the nform (n = dimX) w A *c is integrable.
Appendices
236
XVI.
Some Results on Noncompact Manifolds.
Theorem. Let w be a square integrable and closed (dw = 0) C?‘differential form on an oriented Riemannian manifold. TheTe exists a square integrable harmonic form wo such that w =
wo+&.
(6 some COOdifferential form.) But in contrast to the compact case the form wg needs not to be unique and not each square integrable harmonic form needs to be closed. But there is a very remarkable Theorem. Let (X,g) each square
integrable
be an oriented complete Riemannian harmonic form is closed.
manifold.
Then
What does complete mean? Let a : [O, l] f
D
be a Cmdifferentiable curve in a domain D c FPnwhich is equipped with a Riemannian metric. The velocity of o at t E [0, l] is defined by G(t)
=
c
sij (+))&(t)~j(t))
*
l
and hence obtain
a natural
c
The

in general is neither injective
theorem
of de Rham
between
the
without
compact
de Rham
states
that
cohomology
support)
with
7
linear mapping qq
which
CP(W
HP(X)
t
nor surjective.
there
are natural
isomorphisms
W(X)
= HP(X,
C)
H,p(X)
Y H,p(X,
C)
groups
and
coefficients
the singular
cohomology
groups
(with
or
in C.
The Poincare duality theorem is usually proved in the context of singular cohomology. We express it in terms of the “de Rham cohomology”. First
we construct
a pairing HP(X)
We represent
x H,nyX)
elements of HP(X)
+
c .
(resp. HFP(X))
by differential
(resp. w’ E Crp(X))
w E Cp(X)
We can consider the nform w A w’. It has compact integrable. We claim that the integral
forms
. support
and is hence
J WAW’ X
depends
only on the class of w or w’. This means for example
JX
o?GAw’=Q.
We have dr;, A w’ = d(3 A w’) and the assertion Stokes’s Theorem. we have
that
Let w be a C”
(n  l)form
follows
from a special case of
with compact support.
JoLJ=o. X
Then
241
III. Alternating Differential Forms So Stokes’s theorem gives us the desired pairing HP(X)
x H,“yX)
+
c I
The Poincark duality theorem states that this pairing is nondegenerate under certain assumptions. (A bilinear mapping vxw+c
(u,b)+Ka,b>
for two vector spaces V, W is called nondegenerate if for each a E V, a # 0, there exists a b E W such that < a, b ># 0 and vice versa. The spaces V and W then have the same dimension.) We now assume that X is contained as an open subspace in a compact topological space x. We assume that the topological space ax
:= xx
(with the induced topology of x) is also equipped with a structure as differentiable manifold. We assume furthermore that each “bo~undary” point a E aX admits an open neighbourhood U(u) and a topological mapping $0: U(u)

b v=
{CEE Rn ( IlLElI< 1) 3% 2 O}
such that
p(U(a)nX)
= vi = (~0
I ~,>o}
and such that the mappings U(u) n x
U(u)rdX

vo )
t
{~ER”~
I (z,O)EV},
induced by cp are diffeomorphisms. (This means that X is the interior of a compact Cwmanifold with boundary.)
Poincar~
Duality.
Under the above assumptions on X we have
1) All the cohomology
groups HP(X)
are finite dimensional. 2) The pairing HP(X)
?
Rxv
x H,nP(X) ) c (w,w’)
cf
w A w’ JX
is nondegenerate.
We especially
dimHP(X)
have
= dimHESP(X)
.
Appendices
242
The proof of this theorem is usually reduced to a corresponding result in algebraic topology via the “de Rham isomorphism”. But is is also possible to give a proof in the context of differential forms. In this connection we mention another long exact sequence which is well known from algebraic topology. Under the same assumptions has a long exact sequence * ... 
q(x)

HP(X)
The nature of a is not important natural ones. Example: compact.
satisfies
as in the Poincard
+
duality
a P H,p+l(x)
Hyax)
in our application.

one
... .
All other mappings
are
Let I? be a discrete subgroup of SL(2, R)n such that (Hn)*/I’ is We assume that I’ has no elliptic fixed points. Then the quotient
the assumptions
of the Poincare
duality
theorem.
But the compactification by the cusps is not a manifold We have to modify this compactification. Recall that close to the cusp 00 the quotient l&/r, with
theorem
,
C > 0. We have a natural
&={eHn
boundary.
looks like
1 iVy>C}
topological
TL/r,{toi
H”/I’
with
mapping
1 QC)XY,
where Y = (2 E Hn 1 Ivy=
q/r,.
This space carries a natural differentiable structure. We have proved that it is compact. Hence we may compactify UC/I’, by adding not only a single point but by adding 00 x Y.
&jr
L, [c~,~]xY.
We may repeat this construction for each cusp class and obtain a realization of H”/I’ as the interior of a manifold with boundary. This shows that the Poincark duality theorem can be applied to H”/I’. The spaces Hn/r, *
‘Exact”
means
that
the image
, ?Icp,
, [c,o~] x Y
of an arrow
equals
the kernel
of the next
one.
III. Alternating
Differential
are homotopy
243
Forms
equivalent.
We obtain
Hyax)
E 6 HP(H”,l?,;) j=l
where
ICI,...,
&h is a set of representatives
We therefore . ..+
obtain
an exact
iY,“(H”/I’)
+
sequence
Hm(H”/I’)
of the cusps. which
is used in Chap.
+ 6 j=l
a All
the arrows
H”+l(Hn/I’) c besides
t
... .
the a’s are obvious
ones.
H”(H”,l,j)
III,
$5
Bibliography
Andreotti, A., Vesenlini, E. 1. Carleman Estimates for the LaplaceBeltrami equation on complex manifolds. Publ. Math., I.H.E.S. 25, 313362 (1965) Ash, A., Mumford, D., Rapoport, M., Tai, Y. 2. Smooth compactification of locally symmetric varieties. Math. Sci. Press, Brookline, Mass. 1975 Baily, W.L. 3. Satake’s compactification of V,l . Amer. J. Math. 80, 348364 (1980) Baily, W.L., Borel, A. 4. Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. 84,442528 (1966) Bassendowski, D. 5. Klassifikation Hilbertscher Modulflilchen zur symmetrischen HurwitzMaa%Erweiterung. Bonner Math. Schriften 163 (1985) Blumenthal, 0. 6. Uber Modulfunktionen von mehreren Veranderlichen. Math. Ann. 56,509548 (1903) and 58497527 (1904) Cox, D., Parry, W. 7. Genera of congruence subgroups in Qquaternion algebras. J. f. d. reine u. angew. Math. 351, 66112 (1984) Cartan, H. 8. Fonctions automorphes. Seminaire No. 10, Paris 1957/58 Deligne, P. 9. Theorie de Hodge. I, II Publ. Math., I.H.E.S. 40, 558 (1971) and 44, 577 (1974) Dennin, J. 10. The genus of subfields of K(pR). Illinois J. of Math. 18, 246264 (1984) Ehlers, F. 11. Eine Klssse komplexer Mannigfaltigkeiten und die Auflijsung einiger isolierter Singula&&en. Math. Ann. 218, 127156 (1975) Freitag, E. 12. Lokale und globale Invarianten der Hilbertschen Modulgruppe. Invent. Math. 17,106, 134 (1972) 13. Uber die Struktur der Funktionenkorper zu hyperabelschen Gruppen. I, II J. f. d. reine u. angew. Math. 247, 97117 (1971) and 254, 116 (1972) 14. Eine Bemerkung zur Theorie der Hilbertschen Modulmannigfaltigkeiten hoher Stufe. Math. Zeitschrift. 171, 2735 (1980) Freitag, E., Kiehl, R. 15. Algebraische Eigenschaften der lokalen Hinge in den Spitzen der Hilbertschen Modulgruppen. Invent. Math. 24, 121148 (1974) van der Geer, G. 16. Hilbert modular forms for the field Q(d). Math. Ann. 233, 163179 (1978)
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Bibliography
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Index
algebraic integer 33, 203 algebraic number field 33, 204 alternating differential form 221 ff. alternating product 222 ample 115 arithmetic genus 121 automorphic form 47 local 114 Betti
number
142.
238
Cartan, criterion of 113 class number 37, 209 codifferentiation 230 coherent sheaf 114 ff. commensurable 35 complex space 113 ff. condition of irreducibility first 31, 89, 115 second 54, 115 cusp 12f., 24ff. boundary point 13 infinity 12, 24ff. cusp class 36 cusp form 47 cusp sector 19, 29 cuspidal cohomology 182,
Dirichlet unit theorem discontinuous 7, 21 discrete 7, 21 discriminant 123,‘205
34, 206
ff.
Eisenstein cohomology 2, 148 ff. Eisenstein series 6Off., 148ff., 158ff. analytic continuation 148ff. space of 64f., 183 elliptic fixed point 8, 30 elliptic matrix 8, 83 elliptic substitution 83 exterior differential 223 EulerPoincarB characteristic 111, 116 factor of automorphy 44 finiteness theorem 66 ff. Fourier expansion 44 fundamental domain 19, 89, 219 fundamental form 229 fundamental set 18, 220 GGtzkyKoecher
184
Dedekind zetafunction 122 de Rham cohomology group 142, 234, 239 with compact support 239 de Rham complex 142, 233ff. de Rham, theorem of 143 desingularisation 117 different 210 differential form 221 ff. holomorphic 227 holomorphic transformation 227 on manifolds 233 transformation 224 dimension formula + Selberg trace formula
principle
51, 114
Hecke summation 151 Hermitean metric 230 Hilbert modular group 1, 32ff. Hilbert polynomial 116 Hodge decomposition 176 Hodge numbers 119, 133, 135, Hodge space 135 universal part 140 Hodge theory 234 complex theory 237 real theory 234 hyperbolic matrix 83 hyperbolic substitution 83 ideal class narrow Kiihler
36, 209 127
property
134, 231
185ff.
250
Index
kernel function Koecher principle + GotzkyKoecher
73
LaplaceBeltrami lattice 22, 205 dual 44
operator
Mobius motion multiplier
230
function 162 229 23, 211
norm 205 of ideals number field parabolic parabolic Petersson Poincark Poincare Poincare
principle
210 204
matrix 13, substitution scalar product duality 179, metric 135, series 58 ff.,
ray class 162 regular 45, 47 Riemann metric rotation factor
83 83, 105 68 240ff. 174 77
Selberg trace formula 73 ff., 79 cocompact case 81f., 89 contribution of cusps 89, 108, 110 contribution of elliptic fixed points 89, 110 error term 80f. main term 79 f. Shimizu Lseries 109, 213 Shimizu’s polynomial 111 singularity 30 Sl(2.R) 5 square integrable cohomology 174 ff., 181, 184 star operator 230 Stokes’s theorem 240 total differential 221 totally positive 23 trace 205 trace formula + Selberg trace formula translation matrix 12, 22, 83 transvection matrix 83 universal cohomology upper halfplane 5
228 9, 89
weight
47
184
E. Freitag, University of Heidelberg; R. Kiehl, University of Mannheim
Eta/e Cohomology and the Weil Conjecture With a Historical
Introduction
by J. A. Dieudonne
Translated from the German manuscript by Betty S. and William C. Waterhouse 1988. XVIII, 317 pp. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 13) Hardcover DM 188, ISBN 3540121757 Contents: Introduction.  The Essentials of Etale Cohomology Theory.  Rationality of Weil cFunctions.  The Monodromy Theory of Lefschetz Pencils.  Del&me’s Proof of the Weil Conjecture. Appendices.  Bibliography.  Subject Index.
This book is concerned with one of the most important developments in algebraic geometry during the last decades. In 1949 Andre Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. He himself proved his conjectures by means of an algebraic theory of abelian varieties in the onevariable case. In 1960 appeared the first chapter of the “Elements de Geometric Algebraique” par A. Grothendieck (en collaboration avec J. Dieudonne). In these “Elements” Grothendieck evolved a new foundation of algebraic geometry with the declared aim to come to a proof of the Weil conjectures by means of a new algebraic cohomology theory. Deligne succeded in proving the Weil conjectures on the basis of Grothendiecks ideas. The aim of this “Ergebnisbericht” is to develop as selfcontained as possible and as short as possible Grothendiecks 1adic cohomology theory including SpringerVerlag Berlin Heidelberg New York London Delignes monodromy theory and to present his original proof of the Weil conjectures. Paris Tokyo Hong Kong
G. van der Geer, University of Amsterdam
Hilbert Modular Sutiaces 1988. IX, 291 pp. 39 figs. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 16) Hardcover DM 148,ISBN 3540176012 Contents: Introduction.  Notations and Conventions Concerning Quadratic Number Fields.  Hilbert’s Modular Group.  Resolution of the Cusp Singularities.  Local Invariants.  Global Invariants.  Modular Curves on Modular Surfaces.  The Cohomology of Hilbert Modular Surfaces.  The Classification of Hilbert Modular Surfaces.  Examples of Hilbert Modular Surfaces.  Humbert Surfaces.  Moduli of Abelian Schemes with Real Multiplication.  The Tate Conjectures for Hilbert Modular Surfaces.  Tables. Bibliography.  List of Notations.  Index. Over the last 15 years important results have been achieved in the field of Hilbeti Modular Varieties. Though the main emphasis of this book is on the geometry of Hilbert modular surfaces, both geometric and arithmetic aspects are treated. An abundance of examples  in fact a whole chapter  completes this competent presentation of the subject. This “Ergebnisbericht” will soon become an indisSpringerVerlag Berlin Heidelberg New York London P ensible tool for graduate students and Paris Tokyo Hong Kong researchers in this field.