Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
Algebraic Groups, Hodge Theory,
and Transcendence
G. WÜSTHOLZ
1. Algebraic groups and transcendence. Let 6? be a connected algebraic
group of dimension n defined over a number field K and g = Lie G its Lie
algebra of invariant derivations, which has a natural structure as a üf-algebra.
Then we denote by G c the Lie group obtained from G by base extension from
K to C. This is a complex Lie group and we have an analytic homomorphism
exp : gc -> G c from g c = 0 ®>K C into G C Given a Lie subalgebra b of 9c we obtain by integration a Lie subgroup B of
G c which need not be closed. We say that B is defined over Q if b is.
It is well known that the algebraic group G is an extension of an abelian
variety A by a linear algebraic group L:
0-+L-+G-+A-+0.
If G is commutative then we can write L, after a base change, as a product
of a power of G a , the additive group, and of G m , the multiplicative group,
L = Gk x Glm. Let us now consider the set B(Q) of Q-valued points on B.
Suppose that there exists an algebraic subgroup HEG with
(i) H defined over Q,
(ii) dim H > 0 ,
(iii) Hc Ç B,
where J?c is the subgroup of G c obtained by base change. Then
0=£H(Q) =
Hc(Q)CBc(Q),
so .Bc(Q) is different from 0. The following theorem shows that the converse is
also true.
THEOREM 1 [Wül]. Bc(Q) / O « / and only if there exists an algebraic
subgroup H of G such that (i), (ii), and (iii) hold.
REMARK. Actually in [Wül] the theorem is only proved for G commutative
but it is not difficult to extend the proof also to the noncommutative case.
Theorem 1 has a very straightforward corollary which proves an old conjecture
of Waldschmidt and is also of some independent interest.
© 1987 International Congress of Mathematicians 1986
476
ALGEBRAIC GROUPS, HODGE THEORY, AND TRANSCENDENCE
477
COROLLARY [Wü2], Let b C g c be a Lie subalgebra of gc generated over
Q by algebraic tangent vectors b, i.e., b E exp _ 1 (G(Q)), and b the smallest Lie
subalgebra defined over Q containing b. Then b is the Lie algebra of an algebraic
subgroup defined over Q.
Theorem 1 is actually a transcendence statement as one can see very easily.
A striking example is Baker's theorem in its noneffective version. For this let
a i , . . . , a n be nonzero algebraic numbers.
COROLLARY 2 [Ba]. If ßi logeai H
\-ßn l o g a n = 0 for algebraic numbers
ßi,...,ßn
not all zero then there exist rational integers bi,...,bn
not all zero
such that
fti log a i +
h bn log an = 0.
PROOF. Let G = G £ . Then 0 C = C n and
exp: z = (z1,...,zn)EQc
r-> (ez\...
,eZn)
EGC.
Now let b be the Lie subalgebra generated by the tangent vector
a= (log at,...,
log an).
Then exp(a) = ( « i , . . . , an) is algebraic. The dimension of b is strictly less than
n since ßiZi +
\- ßnzn = 0 defines a proper Lie subalgebra containing b.
Corollary 1 implies now that b is algebraic and therefore the Lie algebra of
an algebraic subgroup H of G of codimension at least one. But it is well known
that algebraic subgroups H of G™ are defined in Lie G™ by linear forms with
integral coefficients. Hence there exists such a linear form bizi H
\- bnzn, not
identically zero, which vanishes on b hence on a. This proves the corollary.
2. Sketch of the proof of Theorem 1. The proof of Theorem 1 is done
by transcendence arguments and is an interplay between analytic, arithmetic,
and algebraic arguments. We divide it in a natural way into a number of steps.
In order to be able to give references and to avoid technical complications we
restrict ourselves to the case that G is commutative.
Step 1. We first embed G into some projective space P ^ . For this we write
G as an extension of an abelian variety A by a product of a torus G ^ and a
unipotent group which we assume to be of the form G^:
0 - • G £ x G'a - • G - • 4 - • 0.
Then we compactify G and obtain a homogeneous space G on which G acts.
This is described in all details in [F-W], which is based on ideas of Serre. G now
is embedded into a projective space P ^ for some N:
G^G
>N
This gives us then the analytic functions we need for the analytic part of the
proof. They are given by
/ = <p o i o exp : Cn ^ LieGc - • P N -
478
G. WUSTHOLZ
These functions have order of growth at most 2. Furthermore if f = (fa,..., fw)
then the functions &• = fi/fo (i = 1,...,N)
satisfy a system of differential
equations, dgj/dzi = hij(g±,.. .,gN) with polynomials hi3(Ti,.. .,TN) for 1 <
i < n and 1 < j < N. These polynomials have coefficients in Q.
Step 2. Let ft,..., ft be coordinates on b ~ C 6 and d/<9ft,... , <9/<9ft the
corresponding derivations in b Ç QC. Let g E G(C) be a complex point with
Xo(g) 7^ 0 and V Ç G a complete algebraic subset (not necessarily reduced);
U Ç PN the open set defined by X0 ^ 0 and V = VnU. Suppose that g E V(G).
Then we say that g is a point of multiplicity at least T on V along B if for all
F E I(V), F = F(X1/Xo,..
.,XN/Xo),
and all 7 = (71, • • • ,7n) € exp^fo)
(d/dç1yi---(d/dçbybFo<poioexP(lii--->ln)
=0
for all 0 < r i , . . . , r*, such that T± +
h r& < T. For simplicity we shall further
assume that b = n — 1. Assume now that g ^ 0 is in Be (Q) and again for
simplicity that g is not a torsion point. We choose the coordinates XQ, ... ,XN
such that Xo(sg) ^ 0 for all s E N . Then one constructs by the so-called
Siegel Lemma for a certain triple of positive real numbers S, T, D » 1 such that
Dn ^> STn~1, a hypersurface X on G of degree at most D that vanishes on the
set {sg, 0 < s < S} to order at least T along J3.
Step 3. By the so-called Schwarz Lemma one shows next that for certain
real numbers S',Tf with S'T171"1 >• Dn the hypersurface constructed in Step 2
vanishes on the set {sg, 0 < s < S'} to order at least T".
•Step 4. The vanishing statement in Step 3 can be transformed into a statement
in linear algebra. It means that the coefficients of the defining equation of X
have to satisfy a system of
f ml
1 „
1 \
> ST'""1
linear equations, and the condition S,T,n~1 » Dn just means that the number
of equations is much larger than the number of coefficients, which is about Dn.
In order to get a contradiction one hopes that the system has maximal rank,
which would imply that all coefficients are zero; in other words X = G, which
contradicts the definition of a hypersurface. Hence the system cannot have maximal rank. By means of some complicated machinery, the so-called "multiplicity
estimates on group varieties" developed in [Ma-Wül], [Ma-Wü2], and [WÜ3],
this statement is translated into statements on the group generated by g and the
analytic subgroup B. The results in [WÜ3] imply that the analytic subgroup
B is degenerate in a certain sense; more precisely they say that B contains an
algebraic subgroup H such that (i)-(iii) hold. This completes the sketch of the
proof of Theorem 1.
3. Hodge theory. Let X be a smooth proper variety of dimension n over
an algebraic number field K. Then there are various ways for associating with
X cohomology theories.
ALGEBRAIC GROUPS, HODGE THEORY, AND TRANSCENDENCE
479
The first one is the singular cohomology. For this let a: K —> C be an
embedding and define Xc '•= X <8>K C via this embedding. Then Xc is a
complex manifold, its singular cohomology Hl(Xci Z) is defined, and one puts
WB(X):=Hi(XcM)'=Hi(Xc,Z)^Q
(t = 0 , l , . . . ) ,
the so-called Betti-cohomology.
The de Rham cohomology is defined as follows. Let Olx,K (i = 0,1,...) be
the sheaf of algebraic differential i-forms. One obtains the complex
^X/K
:
Ox/K - • ftx/K "^ ftx/K "* " ' ' '
Then the algebraic de Rham cohomology is defined to be
#DRP0
S
where H*(X, ^X/K)
* ^
ne
:= H
*(^>^X/x)>
hypercohomology defined in turn as follows. Let
o* ,
—• 7*»*
ll
^
X/K
J
be a resolution of 0 ^ / K by sheaves of injective Ox-modules. Take its associated
simple complex X"* defined as Kl = ®a+b=i Ja,b> Then
Ji*(X,n*x/K):=H*(X,r(X,K*))Then there is the Hodge-cohomology H^odge(X)
tfLdge(X) := 0
Ha(X,Üx/K)
defined as
(i = 0,1,...)-
a+b=i
The two cohomologies HjyK(X) and H^oAge(X) a r e related by the usual spectral
sequence
EÇ>h =
H*(X,(rx/K)=*H&b(X)
and defines a filtration on H^R(X),
the Hodge filtration. Now a well-known
result says that
H*(X,C)^HÎ>R(X)®KC.
Using this isomorphism one defines the periods as follows: Let e1,...,eNi
E
Hl(X, Z) (i = 0,1,...) be a basis of H%(X, C) and e\,..., ejsji the dual basis in
Hi(X, Z). Furthermore let uj\,..., w^,- £ HJùR(X) (Z = 0,1,...) be a basis for
HhR(X). Then
Ni
wjfc = ^2^k,iel
(k =
l,...,Ni).
The coefficients Ukj are called the periods (of weight i) and can be expressed in
the usual way; namely, by integrating UJ^ along er E Hi(X, Z) one finds
,
Ni
ç
Ni
^er
i-\
Jer
i=1
So the complex numbers uj^i are indeed what are classically known to be periods.
480
G. WUSTHOLZ
The vector spaces H%(X, C) carry a so-called Hodge structure, i.e., they admit
a canonical decomposition if*(X,C) = © a + ò = i i f a ' 6 such that Ha>b = Hb>a.
They are said to be of weight i.
EXAMPLE 1. We consider an elliptic curve
E: y' = 4aT - g^x - g3,
g2, gs E K.
Then Eç* is the complex torus G/L for a lattice L of rank 2 that generates C
over the reals. Hence H\(X,Z) = Z71 © Z^- The de Rham cohomology is
obtained as follows. Let LJ = dx/y, r\ — xdx/y. Then a; is a differential of the
first and r\ one of the second kind. They generate H^nfó) over K. The first
one is everywhere holomorphic and therefore in H°(E,VLlE,K). The second one
has two poles without residues. The periods are
Wi = / w,
Vi=
V
(» = 1,2),
and L can be taken to be the lattice generated by u)± and CAREMARK 1. It is not necessary to restrict ourselves to proper schemes. If X
is arbitrary the theory of "logarithmic differentials" leads also to an analogous
theory which includes differentials of the third kind.
REMARK 2. Everything extends to so-called variations of Hodge structures
which arise as follows. Let S be a smooth scheme over C and / : I - > S a proper
and smooth morphism. Then one has to replace the functors if* by the higher
direct image functors Rzf* and obtains so-called variations of Hodge structures.
REMARK 3. One often finds the situation that a Hodge structure of weight i
is isomorphic to a Hodge structure coming from a Hodge structure of weight
j for some j . This is where one has to introduce the so-called Tate-twists
Q B ( 1 ) , Q D R ( 1 ) defined as
QB(1) = 2™Q,
QDR(1)=#
and for integers m
H&X)(m)
= H£(X) ® Q B ( l ) ® m ,
4 ( X ) ( m ) = H^R(X)
and writes Q B ( ^ ) = Qß(l)® m ; and the same for QDR(Wï)Hodge structure of weight —2, and its Hodge decomposition is
® QB(l)®m
QB(1)
defines a
Q B ( 1 ) ® G = if" 2 ' 0 © J5T 1 «- 1 © if 0 '" 2
with if" 2 ' 0 = H°~2 = 0; hence Q B ( l ) 0 C = H'1'1.
The comparison isomorphism relates the algebraic de Rham cohomology with the analytic Betti
cohomology. This interplay between analytic and algebraic leads to a transcendence problem.
CONJECTURE. Let X be a smooth proper scheme of dimension n over K.
Then the periods of X of weight i are either zero or transcendental.
For i = 1 this conjecture is due to Grothendieck [Gr]. The following theorem
solves this conjecture of Grothendieck.
ALGEBRAIC GROUPS, HODGE THEORY, AND TRANSCENDENCE
481
THEOREM 2. Let X be a smooth proper scheme over K of dimension n.
Then the periods of weight 1 are either zero of transcendental.
The proof of this theorem uses Theorem 1 and will be published in the Mathematische Annalen [WÜ4].
In many situations one can reduce the study of the periods of higher weight
to those of weight one.
EXAMPLE 2. Let n = 2m + 1 > 3, let n, m be integral and for a = (a\,..., ar)
with integers ai > 1 let V^a) Ç p n + r be a complete intersection of dimension
n of hypersurfaces of degree a\,..., ar of "niveau de Hodge 1." This means that
(i) H<*(X,npx)=Oïorp + q = n, \q-p\ > 1,
(ii) Hm(X, n ™ + 1 ) ^ o .
The list of possible V ^ a ) can be found in [Ra]. Then there exists an isomorphism
of polarized Hodge structures
a: H£(X)(m) ^
H&(J(X)),
where J(X) is the so-called intermediate Jacobian of X and is defined as follows.
It is the complex torus
J(X) = Hn(X, Z) \ Hn(X, C ) / i f m + 1 ' m ,
where Hn(X,C)
= Hm^^n © if™.™*1 is the Hodge decomposition.
m m+1
J(X) = i f '
/ i f n ( X , Z ) . One obtains then the following corollary.
Hence
COROLLARY. The periods of the Hodge structure ifg(X)(m) are either zero
or transcendental.
EXAMPLE 3. Let X be an algebraic ff3-surface over K. Then again one
associates with H#(X)(1) via the Clifford algebra constructed from the primitive
cohomology and the intersection pairing on X a certain abelian variety and again
one can apply Theorem 2.
4. Vector spaces associated with rational integrals. We consider a
smooth quasiprojective variety X defined over a number field K and having a
K-rational point, and w E H°(X,QX,K)
a holomorphic differential form on X.
We are interested in the periods / w, 7 G H\(X,Z).
The link between these
periods and Theorem 1 is given by the following result which generalizes a result
of Serre [Se].
THEOREM 3 [F-W]. Letcj E T(X,Ü1X) be a holomorphic differential form
with du) = 0. Then there exists a commutative algebraic group G over K, a
morphism ip: X —• G, and an a E g* = T(G, QQ)00115*" , Q := Lie G, such that
w — (p*a.
The proof of this theorem uses the construction of the so-called generalized
Albanese variety. If one combines this result and Theorem 1 one obtains the
following theorem.
482
G. WUSTHOLZ
THEOREM 4 [WÜ4]. The numbers fu
transcendental.
(7 G ifi(X,Z)) are either zero or
It is a natural question to look at vector spaces generated by numbers / u (7 G
i f i ( X , Z ) , u E if 0 ( J ^ f i ^ ) ) . Since in general i f ° ( X , f t ^ / K ) is not finitedimensional, one has to look at canonical finite-dimensional subspaces of this
space.
Special cases of questions of this type have been treated by various authors (Siegel, Schneider, Baker, Coates, Masser). But they all dealt with lowdimensional algebraic groups. Till now we have not been able to work out a
general result on this question depending intrinsically on the cohomology of X.
But in important special cases we can give explicit results. For this let 7 b e a
smooth projective variety and D C Y a n ample divisor, both defined over some
number field K. Then put X = Y \ D. In this case we have a surjection
n:H°(X,iix/K)^HhR(Y)
by a result of Hodge and Atiyah [H-A]. Therefore we can find a basis for HnR(Y)
(up to exact forms) from forms in H°(X, ftx/K)- We take therefore a section of
7T, i.e., a subspace V of H°(X, iix/K)- Then we denote by V(X) the vector space
generated by 1, 2TTì, fu (7 G H±(X, Z), u E V). Let Alb(Y) be the Albanese
variety of Y. Then up to isogeny we have
Alb(Y) = A? 1 x • • • x Alk
for pairwise nonisogenous simple abelian varieties Ai,...,Afc defined over K.
Then we have the following result.
THEOREM 5 [WÜ4]. We have
1-
T^
«
( d i m A^) 2
,v^
dim F = 2 + 4 V .} ,_ ,\
^f
dim(EndAi) v
One can for example apply this result in the case that y is a Fermât curve.
EXAMPLE 4. Let X be the Fermât curve of degree N given by
where N > 3. Then a basis for H^R(X)
coordinates)
0Jatb = xNa-lyNb-N
is given by the differentials (x,y affine
dx
(first
kind)
and
T,atb = xN-Na-ly-Nb
dy
(second
kind
)
where a and 6 are rational numbers satisfying 0 < a, b < 1, a+b < 1, Na, Nb E Z.
The periods are given by numbers of the form
/ Wa,ò = c(a,b)B(a,b),
/ r/a>6 = c'(a,b)B(l
-a,l-b)
ALGEBRAIC GROUPS, HODGE THEORY, AND TRANSCENDENCE
483
where B(x,y) = r(x)T(y)/T(x + y) when defined and c(a, b),cf(a, b) E Q. It
is possible to decompose explicitly the Jacobian J(N) of this Fermât curve and
to obtain the dimension of the vector space generated by the numbers l,27ri,
B(a,b), B(l — a, 1 — b) for the given range of a, ft. In particular one obtains the
following corollary.
COROLLARY. For rational numbers a,b with 0 < a,b < 1, a + b < 1 the
numbers l,ir,B(a,b) = T(a)T(b)/F(a + b), ir/B(a,b) are Q-linearly independent.
As an application of this corollary to a problem of Lang let X be a smooth
projective curve over Q of genus g > 1. Then the universal covering space is
the open unit disc B\. Let x E X(Q) be a Q-rational point and (pi : B\ —• Xc
a covering map with <pi(0) = x. After a homothety of B± into Br for some
r G C x one can get a new covering map (pr: Br —• Xc such that <pr(0) = x and
^(0)eQ.
THEOREM 6 [W-W]. LetTi be a torsion free Fuchsian group of finite index
in a triangular group Ai C Aut B\ with signature (p,q, s) and 0 an elliptic flux
point of order p > 1 of A i , X = Ti \ JBi. Then r is transcendental.
REFERENCES
[Ba] A. Baker, Linear forma in logarithms of algebraic numbers. I, Mathematika 13
(1966), 204-216.
[F-W] G. Faltings and G. Wüstholz, Einbettungen kommutativer algebraischer
Gruppen
and einige ihrer Eigenschaften, J. Reine Angew. Math. 354 (1984), 175-205.
[Gr] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes
Études Sci. Pubi. Math. 29 (1966), 95-103.
[H-A] W. V. D. Hodge and M. F. Atiyah, Integrals of the second kind on an algebraic
variety, Ann. of Math. (2) 62 (1955), 56-91.
[ M a - W ü l ] D. W. Masser and G. Wüstholz, Zero estimates on group varieties. I, Invent.
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[Ma-Wü2]
, Zero estimates on group varieties. II, Invent. Math. 80 (1985), 233-267.
[Ra] M. Rappoport, Complément a l'article de P. Deligne "La conjecture de Weil pour
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[Se] J.-P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959.
[W-W] J. Wolfart and G. Wüstholz, Der Uberlagerungsradius gewisser
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Kurven und die Werte der Betafunktion an rationalen Stellen, Math. Ann. 273 (1985),
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[Wül] G. Wüstholz, Algebraische Punkte auf analytischen Untergruppen
algebraischer
Gruppen, Ann. of Math. (2) (to appear).
[Wü2]
, Some remarks on a conjecture of Waldschmidt, Approximations Diophantiennes et Nombres Transcendants, Progr. Math., vol. 31, Birkhäuser, 1983, pp. 329-336.
[Wü3]
, Multiplicity estimates on group varieties, Ann. of Math. (2) (to appear).
[WÜ4]
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B E R G I S C H E U N I V E R S I T ä T G E S A M T H O C H S C H U L E W U P P E R T A L , 5600 W U P P E R T A L 1,
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