Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety

32 Pages·2012·0.21 MB·English

Model theory, geometry and arithmetic of the
universal cover of a semi-abelian variety
B. Zilber,
University of Oxford
July 31, 2003
latest misprint corrections 24.10.05
1 Introduction
I believe that it is a common feeling among experts that nowadays model
theory establishes itself more and more as a universal language of mathe-
matics. \Universal" might be not quite a right word here as very few people
outside logic speak this language, but surely its system of notions and ideas
developed on a very high level of abstraction is proving to have a power to
see many flelds of mathematics in a new and unifying way. In many cases
this new angle of view yields new results but sometimes even a new inter-
pretation itself might be a good cause for research. The present paper is
pursuing rather the latter goal.
We study the L -theory of universal covers of semi-abelian varieties
!1;!
over algebraically closed flelds of characteristic 0; in fact over the complex
numbers C: Slightly simplifying and extending the deflnition, by a semi-
abelian variety over C we mean an algebraic group A(C) (we write the group
multiplicatively) such that its universal cover is Cd; d = dimA: This assumes
that there is an exact sequence
0 ¡! ⁄ ¡!i Cd ¡ex!p A(C) ¡! 1; (1)
where exp is an analytic homomorphism from the additive group (Cd;+) and
⁄ = kerexp is a discrete Zariski dense subgroup of Cd isomorphic to ZN; for
1
some N = N ; d • N • 2d: It follows immediately that the torsion in A can
A
be described uniquely by N :
A
Fact 1 Given a semi-abelian variety A and an algebraically closed fleld F
containing the fleld of deflnition of A; for any n the group
A = fa 2 A(F) : an = 1g
n
is isomorphic to (Z=nZ)N:
We are going to discuss the following Uniqueness Problem for covers of
semi-Abelian varieties.
Let A be a semi-abelian variety deflned over some k ; a flnitely generated
0
extension of Q; let V be an abelian divisible torsion-free group and ex an
V
abstract group homomorphism such that
0 ¡! ZN ¡i!V V ¡ex!V A(C) ¡! 1 (2)
is an exact sequence.
Uniqueness Problem Does there exist an isomorphism between the se-
quences (1) and (2), that is a pair of bijections (‰;…) such that ‰ : Cd ¡! V
is a group isomorphism and … : A(C) ¡! A(C) is a bijection induced by a
fleld isomorphism flxing k (a Galois automorphism over k ), and the dia-
0 0
gram commutes?
0 ¡¡¡! ⁄ ¡¡¡i! Cd ¡¡ex¡p! A(C) ¡¡¡! 1
‰ …
? ? ?
0 ¡¡¡! Z?yN ¡¡i¡V! V?y ¡¡ex¡V! A(?yC) ¡¡¡! 1
Notice that the positive answer to the question would signal that (2) is a
reasonable ’algebraic’ substitute for the classical complex universal cover.
This, in turn, could be extended to suggest an algebraic substitute for uni-
versal covers for semi-abelian varieties over flelds of positive characteristic,
replacing ZN by a suitable flnite rank subgroup, e.g. for A equal to a one-
dimensional algebraic torus, the kernel of ex in characteristic p has to be
V
the additive group
1 m
Z[ ] = : m;k 2 Z;k ‚ 0 :
p ‰pk (cid:190)
2
We studied the Uniqueness Problem in [Z0] for A(C) = C⁄; the mul-
tiplicative group of the complexes, that is of the complex one-dimensional
torus, and managed to answer the question positively with the help of some
fleld arithmetic results as well as some quite advanced model theory. Notice
that if we require … to be identity the answer is negative even for this simple
case.
The uniqueness problem remains open even for the cases where we believe
theanswerispositive,e.g. ellipticcurveswithoutcomplexmultiplication,but
itisratherclearthatthesecanbesolvedinpositiveprovidedtheobviousgen-
eralisations of the arithmetic results of [Z0] can be proved. In this paper we
show the converse, that is, in order for the answer to the problem to be posi-
tive generalisations of the arithmetic results used in [Z0] must hold. In other
words, the geometrically motivated Uniqueness Problem in a rather non-
trivial way is equivalent to some profoundly arithmetical questions. The link
between arithmetic and model theory is provided by deep results of J.Keisler
[K] and S.Shelah [Sh] after an observation that the uniqueness problem can
be reformulated as a problem on categoricity in uncountable cardinals of an
appropriate L -sentence. In section 5 we give a list of arithmetic proper-
!1;!
ties which are necessary and su–cient for the sentence to be categorical in
all uncountable cardinals.
The criterion, as remarked above, holds for some classes but it does not
cover the general case. In particular, Theorem 1 of this paper states that a
necessary condition for the existence of the isomorphism is that the action of
the Galois group Gal(k~ : k ) on the Tate module T (A) is represented by a
0 0 l
subgroup of GL (Z ) of flnite index. This is true for elliptic curves without
N l
complex multiplication by a result of Serre, but is false e.g. if the elliptic
curve has a complex multiplication.
A more appropriate version of the Uniqueness Problem assumes the in-
troduction of a more sophisticated structure on V; which yet should not be
too complicated. An expanded version may bear a structure of a module
of complex multiplications on V as well as, say, a bilinear form on ⁄: This
choice is restricted by the model-theoretic criterion on keeping the structure
analysable (preferably stable) and on the other hand we want the analysis to
cover a wider class of arithmetic examples. We would like to address these
matters in a further research.
The results of this paper were conjectured by the author in a vague
form after the main result of [Z0] was obtained. The author is grateful to
3
E.Hrushovski for a suggestive discussion of the topic. Thanks are also due to
O.Lessmann for his educating lectures on Keisler-Shelah theory of excellency
and many helpful discussions. My special thanks to the anonymous referee
who suggested a number of important improvements to the paper.
Misha Gavrilovich achieved some progress in the solution of the Unique-
ness Problem for elliptic curves without complex multiplication, and discus-
sions with him not only were useful but also substantially in(cid:176)uenced the flnal
form of results of section 5.
2 The flrst order theory of group covers
We consider a natural language of two sorted structures V to describe the
universal covers. The flrst sort, denoted usually V; corresponding to Cd; is
going to be a group structure in the language (+;q¢) ; which treats V as
q2Q
a rational vector space.
Thesecondsortdescribesthealgebraic groupAasagrouponthesetA =
A(F) of F-points of A, for some algebraically closed fleld F of characteristic
zero (which is just C in the initial setting). Such a group can be represented
as a constructible (Boolean combination of Zariski closed) subset A(F) (cid:181)
Pn(F) of the projective space over F; with an algebraic group operation. Let
k = Q(c) be a fleld which contains the fleld of deflnition of A; c a flnite tuple
0
from F: We consider all Zariski closed k -deflnable relations W (cid:181) An on A as
0
part of the language, that is each of the relations is named in the language.
Notice that the group operation corresponds to one of the relations. So, from
now on when we refer to A as a substructure of V we have all the Zariski
closed relations over k on A in mind. We now refer to a well-known
0
Fact 2 (Folkloreand[Z1], [Z2])InA(F)analgebraicallyclosedfleld(F(A);+;¢)
is deflnable. Moreover, if we choose c in the deflnition of k big enough,
0
A(F) (cid:181) dcl F(A) and F(A) (cid:181) dcl (A(F));
or equivalently: for any point a 2 A there is a flnite tuple [a] in F such
that any automorphism of the structure that induces identity on [a] acts as
identity on a and vice versa.
Corollary 1 We can identify the initial fleld F with F(A) and A(F) with
the A:
4
Remark Technically, the identiflcations can be realised via a flnite collection
of meromorphic functions f ;:::;f on A such that
1 n
f (a) = f (a0) &::: &f (a) = f (a0) ifi a = a0; for generic a;a0 2 A:
1 1 n n
Then for such an a one can let [a] = hf (a);:::;f (a)i:
1 n
To complete the description of V we indicate that one more operation
ex : V ! A acts between the two sorts.
The flrst-order axioms for group covers of a flxed semi-abelian variety A
say:
A1. (V;+;q¢) is a Q-vector space;
q2Q
A2. The complete flrst order theory of A(F) in the relational language
having a name for each algebraic variety W (cid:181) An deflned over k = Q(c);
0
A3. ex is a group homomorphism from (V;+) onto the group (A(F);¢):
We let T be the flrst order theory axiomatised by A1 - A3.
A
It follows from the uncountable categoricity of the theory of algebraically
closed flelds of flxed characteristic and Fact 2
Fact 3 GivenT ;anuncountablecardinal•andamodelofT withcard F =
A A
•; the isomorphism type of the structure on A(F) described by axioms A2 is
determined uniquely.
In other words, if there is another model with card F0 = •; then there
is an isomorphism … : A(F) ! A(F0) of the substructures inducing a fleld
isomorphism F ! F0 over k :
0
Moreover, the theory of A(F) has elimination of quantiflers in the lan-
guage of Zariski closed relations.
In what follows we usually denote V = (V;A); with A = A(F); models of
T :
A
Given a subgroup S (cid:181) V we write S › Q for the divisible hull of the
subgroup. Also we denote ⁄(V) the kernel of ex in V (which is deflnable by
5
the quantifler-free formula ex(x) = 1) and often we omit mentioning V when
no ambiguity can arise.
Lemma 2.1 T implies that
A
V »= V +_ ⁄(V)›Q; (3)
0
with V a linear subspace, and
0
⁄(V)=n⁄(V) »= (Z=nZ)N; N = N : (4)
A
Proof The flrst follows from the general theory of linear spaces, since
⁄(V)›Q is a subspace. It follows also from the axioms that
A(F) »= V £(⁄(V)›Q)=⁄(V):
0
The second component of the decomposition is isomorphic to the torsion
subgroup of A(F); which is described in Fact 1, and the description is flrst
2
order. Hence (4) follows.
We say that the kernel in V is standard if
⁄(V) »= ZN:
3 Types and elimination of quantiflers
We write the group operation in A multiplicatively.
Let W (cid:181) An be an algebraic variety deflned and irreducible over some
fleld K ¶ k0: With any such W and K we associate a sequence fW 1l : l 2 Ng
of algebraic varieties which are deflnable and irreducible over K and satisfy
the following:
W1 = W; and for any l;m 2 N the mapping
[m] : hy ;:::y i 7! hym;:::ymi
1 n 1 n
1 1
maps Wlm onto W l:
6
Such a sequence is said to be a sequence associated with W over K:
Also with any hw ;:::w i 2 W as above we associate a sequence
1 n
1
fhw1;:::wnil : l 2 Ng
such that for any l;m 2 N the mapping
[m] : hy ;:::y i 7! hym;:::ymi
1 n 1 n
1 1
maps hw1;:::wnilm onto hw1;:::wnil: Such a sequence is said to be associ-
ated with w„ = hw ;:::w i:
1 n
1
A sequence associated with w„ is not uniquely determined; for w„l there
are lNn possible values. Obviously, one can get all the values multiplying a
1 „
value w„l by all the » = h»1;:::;»ni; with »i’s torsion points of order l; which
we sometimes denote „11l: We say that other possible choices of the sequence
associated with the same w„ are conjugated to the given one. The same is
applied to sequences associated with a variety W:
Lemma 3.1 Let w„ 2 W and fw„1l : l 2 Ng; fW1l : l 2 Ng be sequences
associated with w„ and W correspondingly. Then there is a sequence f„11l : l 2
Ng of torsion points associated with „1 = h1;:::;1i 2 An such that
„11l ¢w„1l 2 W1l
for all l 2 N: Moreover, if for every l there is z 2 F such that
l
1 1 1
hw1l;:::;wnl¡1;zli 2 W l;
then we may assume
„11l = h1;:::;1;11li; for some associated sequence f11l : l 2 Ng:
2
Proof Immediate from the deflnitions.
7
Lemma 3.2 Assume that ⁄ in V is algebraically compact (which is the case
if V is !-saturated), W a nonempty algebraic subvariety of Fn and fW1l :
l 2 Ng a sequence associated with W over k: Then there is x„ 2 Vn such that
1
1
ex( ¢x„) 2 W l
l
for all l 2 N: In fact, given any v„ = hv ;:::;v i such that ex(v„) 2 W; we can
1 n
get the required x„ in the form x„ = hv +¿ ;:::;v +¿ i for some ¿ ;:::;¿ 2 ⁄:
1 1 n n 1 n
Moreover, if for every l there is z 2 A such that
l
v v
1 n¡1 1
hex( );:::;ex( );zli 2 W l
l l
then we may assume ¿ = ¢¢¢ = ¿ = 0:
1 n¡1
Proof By 3.1 we need to choose ¿„ such that
¿„
ex( ) = „11l for all l 2 N:
l
This deflnes a consistent type in ⁄ in terms of group operation, and we are
2
done by algebraic compactness.
Lemma 3.3 Given a flnitelygeneratedextensionk of k and v„ 2 Vn; linearly
0
independent, the quantifler-free type of v„over k is determined by the following
three sets of formulas:
1
1
ex( ¢x„) 2 W l : l 2 N ; (5)
‰ l (cid:190)
fex(x„) 2= V : V ‰ W; k-variety; dimV < dimWg; (6)
fm ¢x +¢¢¢+m ¢x 6= 0 : hm ;:::;m i 2 Zn nf„0gg; (7)
1 1 n n 1 n
1
for W the minimal k-variety containing ex(v„) and a sequence W l associated
with the variety.
8
Proof Check all atomic formulas in the type of v„ :
Any atomic formula containing a term with ex is equivalent to a Zariski
closedrelationbetweenex(xi);i =;1:::;n;withacommonl:Thisisincluded
l
in(5). Thenegationofsuchanatomicformulafollowsfrom(6). And(7)lists
all the negations of atomic formulas, which do not contain terms with ex:
2
Positiveatomicformulaswithnoextermscannotholdbytheassumptions.
Remark If dimW = 0 the part given by (6) is void.
Lemma 3.4 Let
V = (V;A) and V0 = (V0;A0)
be !-saturated models of T and
A
‰ : (V [A) ! (V0 [A0)
a partial L-isomorphism, with flnitely generated domain D: Then given any
z 2 V [A; ‰ extends to the substructure generated by D[fzg:
Proof By deflnition the V-part of D is a linear subspace generated by some
linearly independent v ;:::;v 2 V:
1 n¡1
First consider the case z 2 A: We may assume that z 2= ex(V \D); for
otherwise z is in D already. Then the quantifler-free type qftp(z=D) of z over
D is determined by the quantifler-free type qftp (z=D\A) of the structure
A
A of z over D \ A; since the only terms over D \ H that may appear in
the atomic formulas concerning z are of the form ex(q¢v); and these can be
replaced by their values in D \A: In this case we can extend ‰ by choosing
a realisation of the type ‰(qftp (z=D \ A)); which is consistent because of
A
the quantifler elimination for A:
Now consider the case when z 2 V n D: Let C be a flnite subset of A
which, along with fv ;:::;v g; generates D: We can replace the fleld k
1 n¡1 0
by its extension k (C) and thus w.l.o.g. assume that D \A is generated by
0
ex(v ;:::;v ) alone.
1 n¡1
Let, for l 2 N; W1l be the minimal algebraic variety over k0 which con-
tains hex(v1);:::;ex(vn¡1);ex(z)i: Obviously, fW1l : l 2 Ng is a sequence
l l l
associated with W: By assumptions on ‰ and the elimination of quantiflers in
A; for every l there is yl 2 A0 such that hex(‰vl1);:::;ex(‰vnl¡1);yli 2 W1l: By
Lemma 3.2 there is vn0 2 V such that hex(‰vl1);:::;ex(‰vnl¡1);ex(vln0 )i 2 W1l
9
for all l 2 N: Letting ‰(z) = v0 and extending to the subspace generated
n
by D \ V [ fzg by linearity, we have by Lemma 3.3 the required partial
2
isomorphism.
Corollary 2 The flrst-order theory T is submodel complete, allows elimi-
A
nation of quantiflers and is complete and superstable.
Corollary 3 The structure induced in V on the sort A is the structure in-
duced by Zariski closed k -deflnable relations only.
0
Elimination of quantiflers also yields
Corollary 4 Given a model V = (V;A) of T ; the decomposition (3) of
A
Lemma 2.1 and elements ¿ ;:::;¿ 2 ⁄(V) such that
1 N
n ¿ +¢¢¢+n ¿ 2 m⁄ ifi g:c:d:(n ;:::;n ) 2 mZ (8)
1 1 N N 1 N
for any n ;:::;n ;m 2 Z; m > 1; let
1 N
V0 = V +Q¿ +¢¢¢+Q¿ :
0 1 N
Then the substructure V0 = (V0;A) of V is a model of T with standard
A
kernel.
Proof Indeed, ex(V0) = A(F); since ex(Q¿ + ¢¢¢ + Q¿ ) contains all the
1 N
m-torsion points of A(F); for all m; by Fact 1, and thus
ex(Q¿ +¢¢¢+Q¿ ) = ex(⁄(V)›Q):
1 N
This proves that V0 is a model of T :
A
Since ⁄(V0) › Q \ V = 0 and Q¿ + ¢¢¢ + Q¿ (cid:181) ⁄(V0) › Q; we have
0 1 N
Q¿ +¢¢¢+Q¿ = ⁄(V0)›Q and thus Z¿ +¢¢¢+Z¿ = ⁄(V0): 2
1 N 1 N
We call an N-tuple h¿ ;:::;¿ i in ⁄(V) with the property (8) a pseudo-
1 N
generating tuple of ⁄(V):
10