322
SOME F U N D A M E N T A L THEOREMS
ABELIAN FUNCTION
By P E T E R
ON
FIELDS
ROQUETTE
1. Introduction
1.1. The 'fundamental theorems* I have in mind are those which are
centred around the celebrated finiteness theorem of Mordell and Weil.
Let us first recall what this finiteness theorem says:
Let A be an abelian variety which is defined over a number field k.
Let Ak be the group of all points of A which are rational in 1c. Then the
finiteness theorem of Mordell and Weil[1] can be stated as follows: Ak,
as a commutative group, is finitely generated.
That means, there are finitely many points ax,...,am in Ak such that
each point a in Ak can be written in the form
m
a = J^n^t
with rational integers %.
There is a generalization of this theorem which is due to Néron[2].
Namely, Néron has proved that the finiteness theorem does not only
hold for a number field as ground field, but also for a much wider class
of fields: In the above theorem, h may be an arbitraryfinitelygenerated field.
In the following, when referring to the finiteness theorem, we shall
always mean this generalization by Néron rather than the original
theorem.
1.2. The purpose of this note is, first, to give another formulation and
at the same time a further generalization of the finiteness theorem which,
as it seems, is just the 'proper' one from the arithmetical point of view
(see the class theorem in § 2.4.) Secondly, we shall speak about some new
principles and results by means of which the finiteness theorem (and its
generalization) can be proved. Since it is impossible in this short note
to deal with the whole proof, we shall therefore restrict ourselves to
those results which lead to the so-called 'weak' finiteness theorem (or
its generalization, the 'weak' class theorem). This 'weak' theorem says
that for an integer n > 1 the factor group Ak\nAk is finite; it is the first
step towards the proof that Ak itself is finitely generated. It seems to
be interesting that this 'weak' theorem is a consequence of theorems
of a eohomologieal nature (see § 3).
THEOREMS ON ABELIAN FUNCTION FIELDS
323
2. The class theorem.
2.1. First we shall give a reformulation of the finiteness theorem,
presenting it as a statement about divisor classes instead of points of an
abelian variety. In doing this we restrict ourselves to the special case
where the abelian variety in question is the jacobian variety of an algebraic curve (this is no essential restriction).
So let T be a projective curve, defined over a finitely generated field k;
assume that V is normal. Let K = k(T) be the function field of V over k;
this is a finitely generated field extension of k of degree of transcendency
1 over k. By the very definition of the jacobian variety A of T[3], the group
Ak is naturally isomorphic to the group G0 of T-divisor classes of degree
0 of K. This group CQ can be described as follows:
We consider the set M = MT of T-prime divisors of K, i.e. of those
prime divisors of K which are trivial on k. We form the corresponding
divisor group D, defined as the free commutative group whose generators
are the elements of M. We assign to each element u =f= 0 of K its divisor
(u) which counts the zeros and poles of u with their respective multiplicity. Thus we get a homomorphism
diKx^D
of the multiplicative group Kx of K into the divisor group D. This
map d, fundamental in arithmetic, is called the divisor map of K with
respect to the set M of prime divisors.
The cokernel of d, i.e. the factor group of I? modulo the image d(Kx),
is called the divisor class group of K with respect to M. We shall denote
it by G or more precisely by CM whenever we want to indicate the prime
divisor set M which we are referring to.
It is well known that the image d(Kx) is contained in the subgroup D 0
of those divisors which are of degree 0; this fact can also be described by
saying that a function u =f= 0 in K has as many zeros as it has poles. It
follows that the degree of a divisor does not actually depend on the
divisor itself but only on its class in the group C. In particular, the
divisor classes of degree 0 form a certain subgroup of G which we shall
denote by G0 or, more precisely, by C0tM.
It is this group C0 which, as already said above, is naturally isomorphic
to the group Ak of rational points of the jacobian variety A of T. Hence
it follows by the finiteness theorem that GQ is finitely generated. But to
say that CQ is finitely generated is the same as to say that G itself is
finitely generated, for the factor group C/CQ is isomorphic to the additive
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PETER ROQUETTE
group of rational integers. Hence we get the following statement as
another formulation of the finiteness theorem :
Let K be a finitely generated field and T be a projective normal curve
which is a model of K (over some subfield k oiK); denote by M the set
of F-prime divisors of K.
Then the corresponding divisor class group CM of K isfinitelygenerated.
2.2* The generalization of this statement which I have in mind says
that the same is true for an arbitrary model of K (not only for a projective
normal curve). Here, the notion of a model of K may be taken either in
the usual sense in algebraic geometry (over some subfield k of K as
constant field) or in the more general 'arithmetic' sense given to it by
NagataC4] (over some Dedekind domain contained in K). For reasons of
brevity, we shall not explain in detail the definition of a model; we only
give the definition of what we call a 'prime divisor model' of K which
will suffice for our purpose.
So let K b& a finitely generated field and M a set of prime divisors
( = valuations) of K. We denote by R the intersection of all the valuation
rings belonging to M. We say that M is an 'affine' prime divisor model
of K, if the following two conditions are satisfied:
(i) each valuation ring belonging to M is a quotient ring of R;
(ii) R is finitely generated either over the rational integers or over
some subfield k of K.
Now we define a prime divisor model of K as a set M of prime divisors
of K which contains an affine prime divisor model M0 such that M — MQ
is finite.
If a prime divisor model M"of K is given, there is a natural divisor
map d = dM oîKx into the corresponding divisor group, and the cokernel
of d is called, exactly as above, th§ divisor class group of K with respect
toM.
Now we can state our theorem as follows:
Glass theorem. Let M be a prime divisor model of a finitely generated
field K. Then the corresponding divisor class group GM of K is finitely
generated.
2.3. It is not very difficult to deduce this class theorem from the
finiteness theorem of Mordell-Weil-Néron[5]. The reason why I stated
it is that in this form its arithmetic significance becomes clear: it gives
us a certain insight into the structure of the divisor class groups of
finitely generated fields. Note that K may very well be an algebraic
THEOREMS ON ABELIAN FUNCTION FIELDS
325
number field; in this case a prime divisor model of K is just a set of prime
divisors which contains all but a finite number of prime divisors of K.
Hence, in this case, the class theorem reduces to the well-known classical
result that the class number is finite[6].
Hence our class theorem may be regarded as the direct generalization of
the classical result that the class number in an algebraic numberfieldis finite.
2.4. As a side remark we would like to mention that there is another
theorem, dual to the class theorem, which concerns the unit group EM
with respect to a prime divisor model M of K. The unit group is defined
dually to the class group as the kernel of the divisor map.
Let us call a prime divisor model M of K an 'absolute' model, if it
contains an affine model M0 whose corresponding integral domain R0 is
'absolutely' finitely generated, i.e. finitely generated over the rational
integers (see condition (ii) above). Then the unit theorem reads as
follows:
Unit theorem. Let M be an absolute prime divisor model of a finitely
generatedfieldK. Then the corresponding unit group EM of K is finitely
generated.
If K is a number field, then this theorem coincides with the classical
unit theorem of Dirichlet. The general unit theorem can be easily
deduced from this classical statementC5], contrary to the situation for
the class theorem.
3. The weak class theorem
3.1. Let i f be a prime divisor model of a finitely generated field K.
Then the weak class theorem states that for any integer n > 1, prime to
the characteristic of K, the factor group GMjnCM is finite. Of course
this is an immediate consequence of the class theorem; however, as
already said in the introduction, the first step towards the proof of the
class theorem is the weak class theorem.
In this § 3 we shall outline a proof of the weak class theorem.
Let r be the arithmetic dimension of the field K. This is defined as the
degree of transcendency of K if the characteristic of K is + 0, and
1 + degree of transcendency of K if the characteristic of K is 0. If r = 0
then K is finite and the weak class theorem is trivial. If r = 1 then K is
either a number field or a function field in one variable over a finite
constant field. In this case the weak class theorem is true by the classical
theorem on the finiteness of the class number[6]. Hence we may assume
r > 1 and use induction on r.
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PETER ROQUETTE
Let k be a subfield of K, algebraically closed in K, such that K is
separable and of degree of transcendency 1 over k. The key for the
induction argument is the following resultC5].
Proposition 1. / / the weak class theorem holds for all models of k, and
also for all models of K over k, then it holds for all models of K.
Hence,using the induction assumption, we see that it suffices to prove
the weak class theorem for a model r of K over k. It is no essential
restriction to assume that Y is projective and normal; let A be the
jacobian variety of Y. As shown in §2.3, the finiteness of CTjnCv is
equivalent to the finiteness of Ak\nAk. Hence we are back at the proof
of the weak finiteness theorem. However, in our present situation we
have the induction assumption that the weak class theorem is already
proved for the ground field k. This will turn out to be important in the
following proof of the weak finiteness theorem.
Let F be the function field of the jacobian variety A over k. Then
A defines a prime divisor model of F over k; let us denote by GA the
corresponding divisor class group of F and by G0A the subgroup of
those divisor classes in GA which are algebraically equivalent to zero.
It is well known that the group Ak is isomorphic to the group C0 A, such
an isomorphism is given by
x -> class oîdx — d for
x € Ak,
where 6 denotes a theta divisor of A, normed so as to be rational in k.
Hence we have to show that G0jAlnC0tA is finite. This will be done, for
an arbitrary abelian variety, in the following theorems.
3.2. According to what we have said above, we consider the following
situation:
k is a finitely generated field for which the weak class theorem is true;
A is an abelian variety defined over k;
F = k(A) is the function field of A over k;
C0 = C0fA is the group of divisor classes of F with respect to A which
are algebraically equivalent to 0;
n is a given integer > 1, prime to the characteristic of k.
In this situation we have to prove
C0lnC0 is finite.
To prove this, it is no essential restriction to assume that all the nth
division points of A are rational in k. Let us assume this, and let us denote
by Gn the automorphism group of F consisting of the translations by the
nth division points of A.
THEOREMS ON ABELIAN FUNCTION FIELDS
327
3.3. The group Gn acts in a natural way on the divisor group D = DA
of F with respect to J., in such a way that the divisor map d = dA is a
G^-homomorphism. Hence Gn acts also on the cokernel of d, which is
the divisor class group G = CA, and on the kernel of d, which is the
multiplicative group kx of the ground field.
Now, general cohomology theory tells us that there is a natural map
of the cohomology groups of Gn in the cokernel G into the cohomology
groups of Gn in the kernel kx, increasing the dimension by two:
This map has proved to be of fundamental importance in various
problems of arithmetic, in particular in class field theory. We shall
consider it in the dimension i = 0.
The zero cohomology group H°(Gn, G) is essentially the group of those
divisor classes which are fixed under Gn. This group contains in particular
the group CQ, since CQ is elementwise fixed under all translations. Hence,
by restriction to CQ we get a map
hn:G0^H*(Gn,kx),
which we call the Hasse map (with respect to n). It is this map which we
are referring to in the following theorems. Note that H2(Gn, kx) is precisely the group of abelian algebras over k with group Gn in the sense
of HasseC7J.
3.4. The first theorem we would like to mention is the following:
Theorem 1. The kernel of hn is equal to nC0.
Thus hn defines an isomorphism of the factor group C0lnCQ into
H*(Gn,kx).
3.5. The next theorem is about the image of hn. Let us choose an
absolute prime divisor model M of the field k in the sense of § 2.4. We
say that a prime p of M is regular with respect to A (or with respect
to F) if the reduction mod# of J. is non-degenerate and an abelian variety
over the residue field k modp. If this is so then p extends uniquely in a
natural way to Fm.
It is known that all but a finite number of primes p of M are regular
with respect to A. After removing the irregular primes we may therefore
assume that all primes of M are regular with respect to A. We then call
M a regular model of k with respect to A.
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PETER ROQUETTE
Theorem 2. If M is regular with respect to A then the image of hn splits
M-divisorially, i.e. it is contained in the kernel of the natural map
H*(Gn,kx)->H*(Gn,DM)
which is induced by the divisor map dM: kx -> DM.
3.6. The next theorem is about the kernel of the map mentioned in
theorem 2. Before stating it let us remember that the factor group
GMjnCM is finite, according to our assumption that the weak class
theorem is true for the ground field k. Furthermore, we know that the
factor group FMjnEM is also finite; see § 2.4.
Theorem 3. As a consequence of the fact that the groups GMjnCM and
BM\nEM are finite, it follows that the kernel of the natural map
H*{Gn,k*)^H\Gn,DM)
is finite too.
For the proof of the theorems 1, 2 and 3 see [9].
Now, taking the theorems 1, 2 and 3 together we see that hn defines
an isomorphism of the group C0lnC0 into a finite group; hence C0lnCQ
is finite too. This proves the weak finiteness theorem.
3.7. Remark. In order to deduce from the weak finiteness theorem the
finiteness theorem itself one has to apply the classical method of infinite
descent together, with the theory of heights of points of abelian varieties.
For details of a simple proof see my forthcoming paper in the Journal
f. d. reine u. angewandte Mathematik.
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