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 mathematics. “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 fields of mathematics in a new and unifying way. In many cases this new angle of view yields new results but sometimes even a new interpretation itself might be a good cause for research. The present paper is pursuing rather the latter goal.

We study the L_{ω_{1},ω}-theory of universal covers of semi-abelian varieties over algebraically closed fields of characteristic 0, in fact over the complex numbers ℂ. Slightly simplifying and extending the definition, by a semiabelian variety over ℂ we mean an algebraic group 𝔸(ℂ) (we write the group multiplicatively) such that its universal cover is ℂ^{d}, d = dim 𝔸. This assumes that there is an exact sequence

0 ⟶ Λ ⟶^{i} ℂ^{d} ⟶^{exp} A(ℂ) ⟶ 1,  (1)

where exp is an analytic homomorphism from the additive group (ℂ^{d}, +) and Λ = ker exp is a discrete Zariski dense subgroup of ℂ^{d} isomorphic to ℤ^{N}, for some N = N_{A}, d ≤ N ≤ 2d. It follows immediately that the torsion in 𝔸 can be described uniquely by N_{A} :

Fact 1 Given a semi-abelian variety 𝔸 and an algebraically closed field F containing the field of definition of 𝔸, for any n the group

𝔸_{n} = {a ∈ 𝔸(F) : a^{n} = 1}

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 defined over some k_{0}, a finitely generated extension of ℚ, let V be an abelian divisible torsion-free group and ex_{V} an abstract group homomorphism such that

0 ⟶ ℤ^{N} ⟶^{i_{V}} V ⟶^{ex_{V}} 𝔸(ℂ) ⟶ 1  (2)

is an exact sequence.

Uniqueness Problem Does there exist an isomorphism between the sequences (1) and (2), that is a pair of bijections (ρ, π) such that ρ : ℂ^{d} ⟶ V is a group isomorphism and π : 𝔸(ℂ) ⟶ 𝔸(ℂ) is a bijection induced by a field isomorphism fixing k_{0} (a Galois automorphism over k_{0}), and the diagram commutes?

0 ⟶ Λ ⟶^{i} ℂ^{d} ⟶^{exp} 𝔸(ℂ) ⟶ 1
↓  ↓^{ρ}  ↓^{π}
0 ⟶ ℤ^{N} ⟶^{i_{V}} V ⟶^{ex_{V}} 𝔸(ℂ) ⟶ 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 universal covers for semi-abelian varieties over fields of positive characteristic, replacing ℤ^{N} by a suitable finite rank subgroup, e.g. for 𝔸 equal to a onedimensional algebraic torus, the kernel of ex_{V} in characteristic p has to be the additive group

ℤ[\frac{1}{p}] = \{\frac{m}{p^{k}} : m, k ∈ ℤ, k ≥ 0\} .

We studied the Uniqueness Problem in [Z0] for 𝔸(ℂ) = ℂ^{∗}, the multiplicative group of the complexes, that is of the complex one-dimensional torus, and managed to answer the question positively with the help of some field arithmetic results as well as some quite advanced model theory. Notice that if we require π to be identity the answer is negative even for this simple case.

The uniqueness problem remains open even for the cases where we believe the answer is positive, e.g. elliptic curves without complex multiplication, but it is rather clear that these can be solved in positive provided the obvious generalisations 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 positive generalisations of the arithmetic results used in [Z0] must hold. In other words, the geometrically motivated Uniqueness Problem in a rather nontrivial 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_{ω_{1},ω}-sentence. In section 5 we give a list of arithmetic properties which are necessary and sufficient 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(k̃_{0} : k_{0}) on the Tate module T_{l}(𝔸) is represented by a subgroup of GL_{N}(ℤ_{l}) of finite index. This is true for elliptic curves without 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 introduction 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 Λ. 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 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 Uniqueness Problem for elliptic curves without complex multiplication, and discussions with him not only were useful but also substantially influenced the final form of results of section 5.

2 The first order theory of group covers

We consider a natural language of two sorted structures 𝕍 to describe the universal covers. The first sort, denoted usually V, corresponding to ℂ^{d}, is going to be a group structure in the language (+, q·)_{q∈ℚ}, which treats V as a rational vector space.

The second sort describes the algebraic group 𝔸 as a group on the set A = 𝔸(F) of F -points of 𝔸, for some algebraically closed field 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) ⊆ P^{n}(F) of the projective space over F, with an algebraic group operation. Let k_{0} = ℚ(c) be a field which contains the field of definition of 𝔸, c a finite tuple from F. We consider all Zariski closed k_{0}-definable relations W ⊆ 𝔸^{n} on A as 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_{0} on 𝔸 in mind. We now refer to a well-known

Fact 2 (Folklore and [Z1], [Z2]) In 𝔸(F) an algebraically closed field (F (𝔸), +, ·) is definable. Moreover, if we choose c in the definition of k_{0} big enough,

𝔸(F) ⊆ dcl F (𝔸) and F (𝔸) ⊆ dcl (𝔸(F)),

or equivalently: for any point a ∈ A there is a finite 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 field F with F (𝔸) and 𝔸(F) with the A.

Remark Technically, the identifications can be realised via a finite collection of meromorphic functions f_{1}, …, f_{n} on 𝔸 such that

f_{1}(a) = f_{1}(a′) & … &f_{n}(a) = f_{n}(a′) iff a = a′, for generic a, a′ ∈ A.

Then for such an a one can let [a] = ⟨f_{1}(a), …, f_{n}(a)⟩.

To complete the description of 𝕍 we indicate that one more operation ex : V → A acts between the two sorts.

The first-order axioms for group covers of a fixed semi-abelian variety 𝔸 say:

A1. (V, +, q·)_{q∈ℚ} is a ℚ-vector space;

A2. The complete first order theory of 𝔸(F) in the relational language having a name for each algebraic variety W ⊆ 𝔸^{n} defined over k_{0} = ℚ(c);

A3. ex is a group homomorphism from (V, +) onto the group (𝔸(F), ·).

We let T_{A} be the first order theory axiomatised by A1 - A3. It follows from the uncountable categoricity of the theory of algebraically closed fields of fixed characteristic and Fact 2

Fact 3 Given T_{A}, an uncountable cardinal κ and a model of T_{A} with card F = κ, 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 F′ = κ, then there is an isomorphism π : 𝔸(F) → 𝔸(F′) of the substructures inducing a field isomorphism F → F′ over k_{0}.

Moreover, the theory of 𝔸(F) has elimination of quantifiers in the language 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 ⊆ V we write S ⊗ ℚ for the divisible hull of the subgroup. Also we denote Λ(V) the kernel of ex in V (which is definable by the quantifier-free formula ex(x) = 1) and often we omit mentioning V when no ambiguity can arise.

Lemma 2.1 T_{A} implies that

V ≅ V_{0} ˙+ Λ(V) ⊗ ℚ,  (3)

with V_{0} a linear subspace, and

Λ(V)/nΛ(V) ≅ (ℤ/nℤ)^{N}, N = N_{A}.  (4)

Proof The first follows from the general theory of linear spaces, since Λ(V) ⊗ ℚ is a subspace. It follows also from the axioms that

𝔸(F) ≅ V_{0} × (Λ(V) ⊗ ℚ)/Λ(V).

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 first order. Hence (4) follows. □

We say that the kernel in 𝕍 is standard if

Λ(V) ≅ ℤ^{N}.

3 Types and elimination of quantifiers

We write the group operation in A multiplicatively.

Let W ⊆ A^{n} be an algebraic variety defined and irreducible over some field K ⊇ k_{0}. With any such W and K we associate a sequence \{W^{\frac{1}{l}} : l ∈ ℕ\} of algebraic varieties which are definable and irreducible over K and satisfy the following:

W^{1} = W, and for any l, m ∈ ℕ the mapping

[m] : ⟨y_{1}, … y_{n}⟩ ↦ ⟨y_{1}^{m}, … y_{n}^{m}⟩

maps W^{\frac{1}{lm}} onto W^{\frac{1}{l}} .

Such a sequence is said to be a sequence associated with W over K.

Also with any ⟨w_{1}, … w_{n}⟩ ∈ W as above we associate a sequence

\{⟨w_{1}, … w_{n}⟩^{\frac{1}{l}} : l ∈ ℕ\}

such that for any l, m ∈ ℕ the mapping

[m] : ⟨y_{1}, … y_{n}⟩ ↦ ⟨y_{1}^{m}, … y_{n}^{m}⟩

maps ⟨w_{1}, … w_{n}⟩^{\frac{1}{lm}} onto ⟨w_{1}, … w_{n}⟩^{\frac{1}{l}} . Such a sequence is said to be associated with w̄ = ⟨w_{1}, … w_{n}⟩.

A sequence associated with w̄ is not uniquely determined; for w̄^{\frac{1}{l}} there are l^{Nn} possible values. Obviously, one can get all the values multiplying a value w̄^{\frac{1}{l}} by all the ξ̄ = ⟨ξ_{1}, …, ξ_{n}⟩, with ξ_{i}’s torsion points of order l, which we sometimes denote ¯1^{\frac{1}{l}} . We say that other possible choices of the sequence associated with the same w̄ are conjugated to the given one. The same is applied to sequences associated with a variety W.

Lemma 3.1 Let w̄ ∈ W and \{w̄^{\frac{1}{l}} : l ∈ ℕ\}, \{W^{\frac{1}{l}} : l ∈ ℕ\} be sequences associated with w̄ and W correspondingly. Then there is a sequence \{¯1^{\frac{1}{l}} : l ∈ ℕ} of torsion points associated with ¯1 = ⟨1, …, 1⟩ ∈ A^{n} such that

¯1^{\frac{1}{l}} · w̄^{\frac{1}{l}} ∈ W^{\frac{1}{l}}

for all l ∈ ℕ. Moreover, if for every l there is z_{l} ∈ F such that

⟨w1^{\frac{1}{l}} , …, wn−1^{\frac{1}{l}}, z_{l}⟩ ∈ W^{\frac{1}{l}} ,

then we may assume

¯1^{\frac{1}{l}} = ⟨1, …, 1, 1^{\frac{1}{l}} ⟩, for some associated sequence \{1^{\frac{1}{l}} : l ∈ ℕ\}.

Proof Immediate from the definitions.□

Lemma 3.2 Assume that Λ in 𝕍 is algebraically compact (which is the case if 𝕍 is ω-saturated), W a nonempty algebraic subvariety of F^{n} and \{W^{\frac{1}{l}} : l ∈ ℕ} a sequence associated with W over k. Then there is x̄ ∈ V^{n} such that

ex(\frac{1}{l} · x̄) ∈ W^{\frac{1}{l}}

for all l ∈ ℕ. In fact, given any v̄ = ⟨v_{1}, …, v_{n}⟩ such that ex(v̄) ∈ W, we can get the required x̄ in the form x̄ = ⟨v_{1}+τ_{1}, …, v_{n}+τ_{n}⟩ for some τ_{1}, …, τ_{n} ∈ Λ. Moreover, if for every l there is z_{l} ∈ A such that

⟨ex(\frac{v_{1}}{l}), …, ex(\frac{v_{n−1}}{l}), z_{l}⟩ ∈ W^{\frac{1}{l}}

then we may assume τ_{1} = ⋯ = τ_{n−1} = 0.

Proof By 3.1 we need to choose τ̄ such that

ex(\frac{τ̄}{l}) = ¯1^{\frac{1}{l}} for all l ∈ ℕ.

This defines a consistent type in Λ in terms of group operation, and we are done by algebraic compactness. □

Lemma 3.3 Given a finitely generated extension k of k_{0} and v̄ ∈ V^{n}, linearly independent, the quantifier-free type of v̄ over k is determined by the following three sets of formulas:

\{ex(\frac{1}{l} · x̄) ∈ W^{\frac{1}{l}} : l ∈ ℕ\};  (5)
{ex(x̄) ∉ V : V ⊂ W, k-variety, dim V < dim W };  (6)
{m_{1} · x_{1} + ⋯ + m_{n} · x_{n} ≠ 0 : ⟨m_{1}, …, m_{n}⟩ ∈ ℤ^{n} \ {¯0}} ,  (7)

for W the minimal k-variety containing ex(v̄) and a sequence W^{\frac{1}{l}} associated with the variety.

Proof Check all atomic formulas in the type of v̄ :

Any atomic formula containing a term with ex is equivalent to a Zariski closed relation between ex(\frac{x_{i}}{l}), i =, 1 …, n, with a common l. This is included in (5). The negation of such an atomic formula follows from (6). And (7) lists all the negations of atomic formulas, which do not contain terms with ex. Positive atomic formulas with no ex terms can not hold by the assumptions.□

Remark If dim W = 0 the part given by (6) is void.

Lemma 3.4 Let

𝕍 = (V, A) and 𝕍′ = (V′, A′)

be ω-saturated models of T_{A} and

ρ : (V ∪ A) → (V′ ∪ A′)

a partial L-isomorphism, with finitely generated domain D. Then given any z ∈ V ∪ A, ρ extends to the substructure generated by D ∪ {z}.

Proof By definition the V -part of D is a linear subspace generated by some linearly independent v_{1}, …, v_{n−1} ∈ V.

First consider the case z ∈ A. We may assume that z ∉ ex(V ∩ D), for otherwise z is in D already. Then the quantifier-free type qftp(z/D) of z over D is determined by the quantifier-free type qftp_{A}(z/D ∩ A) of the structure 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 ρ by choosing a realisation of the type ρ(qftp_{A}(z/D ∩ A)), which is consistent because of the quantifier elimination for 𝔸.

Now consider the case when z ∈ V \ D. Let C be a finite subset of A which, along with {v_{1}, …, v_{n−1}}, generates D. We can replace the field k_{0} by its extension k_{0}(C) and thus w.l.o.g. assume that D ∩ A is generated by ex(v_{1}, …, v_{n−1}) alone.

Let, for l ∈ ℕ, W^{\frac{1}{l}} be the minimal algebraic variety over k_{0} which contains ⟨ex(\frac{v_{1}}{l}), …, ex(\frac{v_{n−1}}{l}), ex(\frac{z}{l})⟩. Obviously, \{W^{\frac{1}{l}} : l ∈ ℕ\} is a sequence associated with W. By assumptions on ρ and the elimination of quantifiers in 𝔸, for every l there is y_{l} ∈ A′ such that ⟨ex(\frac{ρv_{1}}{l}), …, ex(\frac{ρv_{n−1}}{l}), y_{l}⟩ ∈ W^{\frac{1}{l}} . By Lemma 3.2 there is v′_{n} ∈ V such that ⟨ex(\frac{ρv_{1}}{l}), …, ex(\frac{ρv_{n−1}}{l}), ex(\frac{v′_{n}}{l})⟩ ∈ W^{\frac{1}{l}} for all l ∈ ℕ. Letting ρ(z) = v′_{n} and extending to the subspace generated by D ∩ V ∪ {z} by linearity, we have by Lemma 3.3 the required partial isomorphism. □

Corollary 2 The first-order theory T_{A} is submodel complete, allows elimination of quantifiers and is complete and superstable.

Corollary 3 The structure induced in 𝕍 on the sort 𝔸 is the structure induced by Zariski closed k_{0}-definable relations only.

Elimination of quantifiers also yields

Corollary 4 Given a model 𝕍 = (V, A) of T_{A}, the decomposition (3) of Lemma 2.1 and elements τ_{1}, …, τ_{N} ∈ Λ(V) such that

n_{1}τ_{1} + ⋯ + n_{N}τ_{N} ∈ mΛ iff g.c.d.(n_{1}, …, n_{N}) ∈ mℤ  (8)

for any n_{1}, …, n_{N}, m ∈ ℤ, m > 1, let

V′ = V_{0} + ℚτ_{1} + ⋯ + ℚτ_{N}.

Then the substructure 𝕍′ = (V′, A) of 𝕍 is a model of T_{A} with standard kernel.

Proof Indeed, ex(V′) = A(F), since ex(ℚτ_{1} + ⋯ + ℚτ_{N}) contains all the m-torsion points of 𝔸(F), for all m, by Fact 1, and thus

ex(ℚτ_{1} + ⋯ + ℚτ_{N}) = ex(Λ(V) ⊗ ℚ).

This proves that 𝕍′ is a model of T_{A}.

Since Λ(V′) ⊗ ℚ ∩ V_{0} = 0 and ℚτ_{1} + ⋯ + ℚτ_{N} ⊆ Λ(V′) ⊗ ℚ, we have ℚτ_{1} + ⋯ + ℚτ_{N} = Λ(V′) ⊗ ℚ and thus ℤτ_{1} + ⋯ + ℤτ_{N} = Λ(V′). □

We call an N-tuple ⟨τ_{1}, …, τ_{N}⟩ in Λ(V) with the property (8) a pseudogenerating tuple of Λ(V).

Lemma 3.5 Let 𝕍 = (V, A) be an ω-saturated model of T_{A}, K a subfield of F, V (K) = Ln(𝔸(K)) = {v ∈ V : ex(v) ∈ 𝔸(K)} and let v = ⟨v_{1}, …, v_{n}⟩ be an n-tuple in V linearly independent over V (K) ⊗ ℚ.

Then there is a model (V′, A′) = 𝕍′ ≺ 𝕍 with standard kernel such that 𝔸(K) ⊆ A′ and 𝕍′ realises the type tp(v/𝔸(K)).

Proof Consider the decomposition of the vector space V into the direct sum

V = V (K) ⊗ ℚ ˙+V_{1}

for V_{1} a linear subspace containing v. We can further decompose

V (K) ⊗ ℚ = Λ(V) ⊗ ℚ ˙+V_{2},

for a linear subspace V_{2}. Thus we have

V = (V_{1} + V_{2}) ˙+Λ(V) ⊗ ℚ.

Choose pseudogenerators τ_{1}, …, τ_{N} in Λ(V) and let

V′ = (V_{1} + V_{2}) + ℚτ_{1} + ⋯ + ℚτ_{N}.

By Corollary 4 we have that 𝕍′ = (V′, A) is a model of T_{A}. The rest follows from elimination of quantifiers and the fact that v is in V′.□

Given a sequence \{W^{\frac{1}{l}} : l ∈ ℕ\} associated with W in n variables over a field K and a type p in n variables we say that the sequence stabilises modulo type p if there is an l ∈ ℕ such that for all m there is only one K-definable variety V with V^{m} = W^{\frac{1}{l}} , such that p(x_{1}, …, x_{n}) and ⟨ex(\frac{x_{1}}{ml}), …, ex(\frac{x_{n}}{ml})⟩ ∈ V is consistent.

An obvious equivalent condition is that

p(x_{1}, …, x_{n})∪\{⟨ex(\frac{x_{1}}{ml}), …, ex(\frac{x_{n}}{ml})⟩ ∈ W^{\frac{1}{lm}} \} ⊨ \{⟨ex(\frac{x_{1}}{k}), …, ex(\frac{x_{n}}{k})⟩ ∈ W^{\frac{1}{k}} : k ∈ ℕ\}.

If p is trivial we omit mentioning the type.

Lemma 3.6 Let U ⊆ 𝔸^{N} be the variety given by equations x_{1} = ⋯ = x_{N} = 1. Let K ⊇ k_{0} be a field and, given a pseudogenerating N-tuple τ, the sequence Uτ^{\frac{1}{n}} , n ∈ ℕ, associated with U over K. Then the number of distinct sequences Uτ^{\frac{1}{n}} over K, for all pseudogenerating τ, is either finite or 2^{ℵ_{0}}.

Proof Notice first that if one defines α_{i} = ex(\frac{τ_{i}}{l}) then Uτ^{\frac{1}{n}} describes the locus of ⟨α_{1}, …, α_{N}⟩ (and so determines the complete type of the tuple over 𝔸(k_{0})) and {α_{1}, …, α_{N}} is a basis over ℤ/nℤ of the free finite module

𝔸_{n} = {a ∈ 𝔸(F) : a^{n} = 1}

(see also Fact 1). Moreover, it follows from definition that an N-tuple τ′ with ex(\frac{τ′}{l}) ∈ U^{\frac{1}{l}} , for all l, pseudogenerates kernel if and only if U^{\frac{1}{n}} determines the type of a generating N-tuple ⟨β_{1}, …, β_{N}⟩ of 𝔸_{n}, for every n.

Since the α’s and β’s above are independent generators of the same group, there is an invertible (ℤ/nℤ)-linear (in multiplicative form) map σ : 𝔸_{n}^{N} → 𝔸_{n}^{N} such that

σ : ⟨α_{1}, …, α_{N}⟩ ↦ ⟨β_{1}, …, β_{N}⟩.

σ is a group automorphism 𝔸_{n}^{N} → 𝔸_{n}^{N} 0-definable in 𝔸, and σ(Uτ^{\frac{1}{n}}) meets Uτ′^{\frac{1}{n}} . Since both are atoms, we have σ(Uτ^{\frac{1}{n}}) = Uτ′^{\frac{1}{n}} .

Letting n = lm we see that the number Dg_{l,m}(U_{τ}) of choices for varieties Uτ′^{\frac{1}{n}} , some pseudogenerating τ̄′, such that (Uτ′^{\frac{1}{n}})^{m} = Uτ^{\frac{1}{l}} depends only on K. In case the sequence \{Uτ^{\frac{1}{l}} : l ∈ ℕ\} stabilises we obviously have that the number of all such sequences is finite.

In the alternative case consider the sequence L = {l_{1}, …, l_{i}, …, } constructed by induction as l_{1} = 1 and l_{i+1} = l_{i} · m for m minimal such that

Dg_{l_{i},m}(U_{τ}) > 1. Now, given an l_{i} ∈ L, for any choice of Uτ′^{\frac{1}{l_{i}}} there are at least

2 choices of Uτ′^{\frac{1}{l_{i+1}}} such that

(Uτ′^{\frac{1}{l_{i+1}}} )^{\frac{l_{i+1}}{l_{i}}} = Uτ′^{\frac{1}{l_{i}}} .

Hence there are 2^{ℵ_{0}} sequences associated with U over K.□

Proposition 1 Let 𝕍 be a model of T_{A} and ⟨τ, v⟩ = ⟨τ_{1}, …, τ_{N}, v_{1}, …, v_{n}⟩ with v_{1}, …, v_{n} ∈ V linearly independent over Λ(V) ⊗ ℚ and ⟨τ_{1}, …, τ_{N}⟩ pseudo-generators of the kernel.

(i) Suppose that the sequence \{Wτ,v^{\frac{1}{l}} : l ∈ ℕ\} associated with \{ex(\frac{τ,v}{l}) : l ∈ ℕ} over k_{0} does not stabilise modulo the type ’x is a pseudogenerating Ntuple’ (see (8)). Then there are 2^{ℵ_{0}} distinct complete types over k_{0} realisable in uncountable models of T_{A} with standard kernel.

(ii) Suppose that K ⊇ k_{0} is a field such that 𝔸_{tors} ⊆ 𝔸(K), v is linearly independent over V (K) ⊗ ℚ and the sequence \{Wv^{\frac{1}{l}} : l ∈ ℕ\} associated with \{ex(\frac{v}{l}) : l ∈ ℕ\} over K does not stabilise. Then there are 2^{ℵ_{0}} distinct complete types over K realisable in uncountable models of T_{A} with standard kernel.

Proof We may assume that 𝕍 is ω-saturated.

(i) Consider first the N-type

p_{τ}(x) = \{ex(\frac{x}{l}) ∈ Uτ^{\frac{1}{l}} : l ∈ ℕ\},

with ex(\frac{τ}{l}) ∈ Uτ^{\frac{1}{l}} for all l ∈ ℕ. In case p_{τ} does not stabilise modulo the type ’x is a pseudogenerating N-tuple’ by Lemma 3.6 we have 2^{ℵ_{0}} distinct types of the form p_{τ} for pseudogenerating τ. By Corollary 4 each such type is realised in a model of T_{A} with standard kernel.

In the opposite case p_{τ}(x) can be replaced by a formula ψ(x) and the type ’x is a pseudogenerating tuple’.

Let, for each l ∈ ℕ,

Zτ,v^{\frac{1}{l}} = \{z ∈ 𝔸^{n} : ⟨ex(\frac{τ}{l}), z⟩ ∈ Wτ,v^{\frac{1}{l}}\}

These varieties can be also represented as the fibres over ex(\frac{τ}{l}) of the

projection of Wτ,v^{\frac{1}{l}} into the N-space on the first N co-ordinates.

Claim. For any l there is m and a_{l,m} ∈ 𝔸_{m}^{n} such that

Zτ,v^{\frac{1}{lm}} ∩ a · Zτ,v^{\frac{1}{lm}} = ∅.

Proof. Since \{Wτ,v^{\frac{1}{l}} : l ∈ ℕ\} does not stabilise, for any given l there is an m and some k_{0}-irreducible W^{\frac{1}{lm}} such that

Wτ,v^{\frac{1}{lm}} ∩ W^{\frac{1}{lm}} = ∅  (9)

and

(Wτ,v^{\frac{1}{lm}})^{m} = Wτ,v^{\frac{1}{l}} = (W^{\frac{1}{lm}})^{m}.  (10)

We can assume that l is big enough in order for both pr(Wτ,v^{\frac{1}{lm}}) and pr(W^{\frac{1}{lm}}) to be equal to the same Uτ^{\frac{1}{l}} , the member of the stabilised sequence above. It follows from (9) that the fibres over same point of the projection of the sets do not intersect. It follows from (10) that the fibres are conjugated by a multiple a of order m. This proves the claim.

Now consider a sequence L = {l_{1}, …, l_{i}, …, } constructed by induction as l_{1} = 1 and l_{i+1} = l_{i} · m for m minimal given by the claim.

Now, given a sequence μ : ℕ → 𝔸_{tors}^{n} with the property

μ(1) = 1 and μ(i + 1)^{l_{i+1}} = μ(i)

we have that μ(m) ∈ 𝔸_{l_{1}...l_{m}}^{n} for all m and we can construct an n-type over k_{0}(𝔸_{tors})

q_{μ,τ}(y) = \{ex(\frac{y}{l_{1} … l_{m}}) ∈ μ(m) · Vτ,v^{\frac{1}{l_{1}...lm}} : m ∈ ℕ\}.

By the claim there are 2^{ℵ_{0}} mutually inconsistent such types.

Notice that, since the N-tuple ex(\frac{τ}{l}) generates the group 𝔸_{l} for all l, μ(m) = M(m, τ) is a term of τ. Now we consider the N +n-types in variables {x_{1}, …, x_{N}, y_{1}, …, y_{n}}

Q_{μ}(x, y) = \{ex(\frac{y}{l_{1} … l_{m}}) ∈ M(m, x) · Vx,v^{\frac{1}{l_{1}...lm}} : m ∈ ℕ\}

obtained by replacing all occurrences of τ in q_{μ,τ} by x. Let

Q_{μ}^{∗}(x, y) = Q_{μ}(x, y)&ψ(x)&{ x is a pseudogenerating tuple}.  (11)

We claim that Q_{μ_{1}}^{∗}(x, y) and Q_{μ_{2}}^{∗}(x, y) are consistent if and only if Q_{μ_{1}}^{∗}(τ, y) and Q_{μ_{2}}^{∗}(τ, y) are. This follows from the fact that ψ(x)& ’x is a pseudogenerating tuple’ is a complete type, by Lemma 3.3, since it is equivalent to p_{τ}. Hence there are 2^{ℵ_{0}} mutually inconsistent types of the form (11).

For each such type there is a realisation of the form ⟨τ, v′⟩ in 𝕍. It follows that v′_{1}, …, v′_{n} are linearly independent over Λ(V) ⊗ ℚ. Hence the linear subspace L = ℚv′_{1} + … ℚv′_{n} does not intersect Λ(V) ⊗ ℚ. Hence we can choose linear subspace V′_{0} ⊇ L of V such that

V′_{0} ˙+ Λ(V) = V.

By Corollary 4, for

V′ = V′_{0} + ℚτ′_{1} + ⋯ + ℚτ′_{N}

𝕍′ = (V′, A) is a model of T_{A} with standard kernel. By elimination of quantifiers the types of ⟨τ, v′⟩ in 𝕍 and 𝕍′ coincide. This finishes the proof of (i).

(ii) Given the K-irreducible variety W = W_{v} ⊆ 𝔸^{n}, any sequence \{W^{\frac{1}{l}} : l ∈ ℕ} associated with W over K has the property that for every l, m ∈ ℕ the number Dg_{l,m}(W, K) of K-varieties X ⊆ 𝔸^{n} such that X^{m} = W^{\frac{1}{l}} depends on l and m but not on the way the associated sequence has been chosen. This follows from the fact that if Z^{\frac{1}{l}} is another choice for the lth member of the sequence then, for some a ∈ 𝔸_{lm}^{n},

W^{\frac{1}{l}} = a^{m} · Z^{\frac{1}{l}}

and the map x ↦ ax sets a K-definable bijection

\{x ∈ 𝔸^{n} : x^{m} ∈ W^{\frac{1}{l}} \} → \{x ∈ 𝔸^{n} : x^{m} ∈ Z^{\frac{1}{l}} \}.

This property allows us to construct under the assumption that \{Wv^{\frac{1}{l}} \} does not stabilise a binary tree of 2^{ℵ_{0}} mutually inconsistent sequences associated with W. For each such sequence \{W^{\frac{1}{l}} \} we can, using Lemma 3.5, construct a model of T_{A} with standard kernel in which the sequence is realised. □

Remark (i) The assumption that v_{1}, …, v_{n} are linearly independent over Λ ⊗ ℚ is equivalent to the fact that the elements ex(v_{1}), …, ex(v_{n}) are multiplicatively independent in the group A, that is no non-trivial group word on the elements is equal to 1.

(ii) The assumption that v_{1}, …, v_{n} are linearly independent over V (K) ⊗ ℚ is equivalent to the fact that the elements ex(v_{1}), …, ex(v_{n}) are multiplicatively independent in the group A over 𝔸(K), that is no non-trivial group word on the elements is in 𝔸(K).

4 The L_{ω_{1},ω}-theory of group covers

We start with Remark There is an L_{ω_{1},ω}-formula stating that Λ ≅ ℤ^{N} :

∃τ_{1}, …, τ_{N} ∈ Λ ⋀_{(z_{1},...z_{N})∈ℤ^{N} \{¯0}}z_{1}τ_{1} + ⋯ + z_{N}τ_{N} ≠ 0∧

∧∀u ∈ Λ ⋁_{(z_{1},...z_{N})∈ℤ^{N}} z_{1}τ_{1} + ⋯ + z_{N}τ_{N} = u.

Now we observe that the Uniqueness Problem as formulated in the introduction (without complex multiplication) is equivalent to the following model theoretic question:

Given a semi-abelian variety 𝔸, is the L_{ω_{1},ω}-sentence T_{A} + {Λ ≅ ℤ^{N}} categorical in power 2^{ℵ_{0}}?

Naturally, there are no model theoretical reasons to distinguish 2^{ℵ_{0}}, except maybe for the case 2^{ℵ_{0}} = ℵ_{1}, so we consider rather two versions of the problem:

Given a semi-abelian variety 𝔸, is the L_{ω_{1},ω}-sentence T_{A} + {Λ ≅ ℤ^{N}} categorical in

(i) power ℵ_{1}?

(ii) all uncountable powers?

In these forms the problem can be treated in the frames of Keisler-Shelah theory of L_{ω_{1},ω}-categoricity.

The first step in this theory is to reduce the study of a categorical sentence to the case when all the models of the sentence are first-order atomic. This is done in general by extending the language by appropriate L_{ω_{1},ω}-definable predicates.

We assume below that T_{A} + {Λ ≅ ℤ^{N}} is ℵ_{1}-categorical.

Notation Let

(i) for each n > 0, Ind^{n}(x_{1}, …, x_{n}) denote an n-type stating that x_{1}, … x_{n} are linearly independent in the ℚ-space V;

(ii) P G^{l}(x_{1}, …, x_{l}), l ∈ N, be the l-type:

⟨x_{1}, …, x_{l}⟩ ∈ Λ^{l} & ⋀_{g.c.d.(m_{1},...,m_{l})=1, m>1} m_{1}x_{1} + ⋯ + m_{l}x_{l} ∉ mΛ;

(iii) for each n and a k_{0}-irreducible variety W in n variables Gen^{W} (ȳ) be the n-type on A stating that ȳ is k_{0}-generic point in W.

Remark A new predicates of type (i) is equivalent to the set of formulas (7) and of (iii) to the set (6).

Remark It is immediate by definitions that an l-tuple ⟨x_{1}, …, x_{l}⟩ can be extended to a pseudo-generating N-tuple iff P G^{l}(x_{1}, …, x_{l}) holds.

In the following definition we use vector and matrix notations: x = (x_{1}, …, x_{n}), and for r = (r_{1}, …, r_{n}) ∈ ℚ^{n}, we denote rx = r_{1}x_{1} + ⋯ + r_{n}x_{n}.

Call a quantifier-free L-type p(x_{1}, …, x_{n}) almost principal if p is equivalent to a union of the following subtypes, for some rational vectors q_{1}, …, q_{m} ∈ ℚ^{n}, (m ≤ n), a rational n × m-matrix Q, a non negative integer l ≤ m, a positive integer M and a k_{0}-irreducible variety W in m variables:

(i) Ind^{m}(q_{1}x, …, q_{m}x) & x = ⟨q_{1}x, …, q_{m}x⟩Q;

(ii) P G^{l}(q_{1}x, …, q_{l}x);

(iii) ⟨ex(q_{1}x), …, ex(q_{m}x)⟩ ∈ W^{\frac{1}{M}} & Gen^{W} (ex(q_{1}x), …, ex(q_{m}x)

Let S_{A} be the set of all complete n-types in V -variables, for all n, realisable in models of T_{A} + {Λ ≅ ℤ^{N}}.

Lemma 4.1 Any type in S_{A} is almost principal or |S_{A}| = 2^{ℵ_{0}}.

Proof Assume that there is a non-almost principal type realised in a model 𝕍 of T_{A} + {Λ ≅ ℤ^{N}} by some ⟨v_{1}, …, v_{n}⟩ ∈ V^{n}.

We claim that then, for τ_{1}, …, τ_{N} generating Λ(V), the type of ⟨τ_{1}, …, τ_{N}, v_{1}, …, v_{n}⟩ is not almost principal as well. Indeed, we may assume w.l.o.g. that τ_{1} = v_{1}, …, τ_{l} = v_{l} and {v_{l+1}, …, v_{n}} is linearly independent over Λ ⊗ ℚ. Let W be the minimal k_{0}-variety containing ⟨exv_{1}, …, exv_{n}⟩. Our assumptions imply that, for no positive integer M, the type

Ind^{n}(v_{1}, …, v_{n}) & P G^{l}(v_{1}, …, v_{l}) & Gen^{W} (ex(v_{1}), …, ex(v_{n})) &
&⟨ex(v_{1}), …, ex(v_{n})⟩ ∈ W^{\frac{1}{M}}

implies

\{⟨ex(\frac{v_{1}}{m}), …, ex(\frac{v_{n}}{m})⟩ ∈ W^{\frac{1}{Mm}} : m ∈ ℕ\}.

This is also true if we replace P G^{l}(v_{1}, …, v_{l}) with P G^{N}(v_{1}, …, v_{l}, u_{l+1}, …, u_{N}), for new variables u_{l+1}, …, u_{N}, since an l-tuple can be extended (in a saturated model of T_{A}) to a pseudogenerating N-tuple iff the l-tuple satisfies P G^{l}.

So, we now are under assumptions of Proposition 1 and hence |S_{A}| = 2^{ℵ_{0}}.□

Keisler’s Theorem (Theorem 5.6 of [K]) If an L_{ω_{1},ω}-sentence Σ is ℵ_{1}- categorical then the set of complete n-types realisable in models of Σ is at most countable.

Corollary 5 All types in S_{A} are almost principal.

Extend the language L to a new language L^{∗} by adding predicate symbols to be interpreted as Ind^{n}(x_{1}, …, x_{n}), P G^{l}(x_{1}, …, x_{l}) and Gen^{W} (ȳ) on A for all n, l, W (l ≤ N).

Remark All the new predicates are L_{ω_{1},ω}-definable in T_{A}.

Recall that an L-structure is said to be L-atomic if the type of any finite tuple of elements in the structure is principal, i.e. is determined by a finite set of L-formulas.

The following is stating that all the uncountable models of T_{A}+{Λ ≅ ℤ^{N}} are ω-homogeneous in the language L^{∗}.

Lemma 4.2 Let

𝕍 = (V, A) and 𝕍′ = (V′, A′)

be models of T_{A} + {Λ ≅ ℤ^{N}} such that the underlying fields F (A) and F (A′) are of infinite transcendence degree. Suppose

ρ : (V ∪ A) → (V′ ∪ A′)

is a partial L-isomorphism with finitely generated domain D. Then, given any z ∈ V ∪ A, ρ extends to the substructure generated by D ∪ {z}.

In particular, if 𝕍 and 𝕍′ are countable, ρ extends to an isomorphism between the structures.

Proof Similarly to the proof of Lemma 2.1.

Let v_{1}, …, v_{n−1} ∈ V generate the linear subspace V ∩ D and be independent.

First consider the case z ∈ A \ ex(V ∩ D). The quantifier-free type qftp(z/D) of z over D is determined by the quantifier-free type qftp_{A}(z/D ∩ A). The latter by QE for 𝔸 is equivalent to a collection of formulas stating that z is generic in W over A ∩ D, for some irreducible over A ∩ D variety W. Then ρ(qftp_{A}(z/D ∩ A)) states the corresponding genericity condition about z′ ∈ ρ(W). Such a z′ must exist in A′ by the assumptions of the lemma. Hence we are done in this case.

Now consider the case when z ∈ V \ D. Let C be a finite subset of A which along with {v_{1}, …, v_{n−1}} generates D. We can replace the field k_{0} by its extension k_{0}(C) and thus w.l.o.g. assume that D ∩ A is generated by ex(v_{1}, …, v_{n−1}) alone.

Let, for l ∈ ℕ, W^{\frac{1}{l}} be the minimal algebraic variety over k_{0} which contains ⟨ex(\frac{v_{1}}{l}), …, ex(\frac{v_{n−1}}{l}), ex(\frac{z}{l})⟩. By Corollary 5 the type

\{⟨ex(\frac{v_{1}}{l}), …, ex(\frac{v_{n−1}}{l}), ex(\frac{z}{l})⟩ ∈ W^{\frac{1}{l}} : l ∈ ℕ\}

is equivalent to its finite subset, in fact just to one of the formulas. By assumptions on ρ and the elimination of quantifiers in 𝔸, given l, there is y_{l} ∈ A′ such that ⟨ex(\frac{ρv_{1}}{l}), …, ex(\frac{ρv_{n−1}}{l}), y_{l}⟩ ∈ W^{\frac{1}{l}} . Then, letting ρ(z) = z′ for a z′ ∈ V′ such that ex(\frac{z′_{1}}{l}) = y_{l}, we get, extending to V ∩ D + ℚz by linearity, the required partial isomorphism. □

Proposition 2 Any model of the L_{ω_{1},ω}-sentence

T_{A} + {Λ ≅ ℤ^{N}} + {tr.d.F (A) ≥ ℵ_{0}}  (12)

is L^{∗}-atomic.

Proof By Lemma 4.2 any complete L^{∗}-type p (in fact any complete L_{ω_{1},ω}- type) in a model of the sentence is determined by its quantifier-free Lsubtype, which is almost principal by Corollary 5. This immediately implies that p is equivalent to a finite L^{∗}-type.□

Keisler’s theory can say more about our Σ. To get stronger consequences notice first

Proposition 3 T_{A} + {Λ ≅ ℤ^{N}} has the amalgamation property, that is for any three models 𝕍_{0}, 𝕍_{1} and 𝕍_{2} of the sentence with embeddings 𝕍_{0}≼_{π_{i}}𝕍_{i}, i = 1, 2 there is a model 𝕍 and embeddings 𝕍_{i}≼_{ϕ_{i}}𝕍, i = 1, 2 agreeing with the π_{i}’s.

Proof Notice that by quantifier elimination the embeddings are just usual embeddings. Let 𝕍_{0} ⊆ 𝕍_{i}, i = 1, 2 be embeddings of models and V_{0} ⊆ V_{i}, 𝔸(F_{0}) ⊆ 𝔸(F_{i}) the corresponding embeddings of the underlying sets. By QE and the definable correspondence between 𝔸(F) we have F_{0} ⊆ F_{i}, i = 1, 2.

Now consider an algebraically closed field F which is a free amalgam of algebraically closed subfields F_{1} and F_{2} over F_{0}, that is, up to isomorphism, we can think of F as acl(F_{1}F_{2}) with F_{1}, F_{2} ⊆ F, F_{0} = F_{1} ∩ F_{2} and transcendence bases B_{i} of F_{i} over F_{0}, i = 1, 2, independent over F_{0}. Let also V_{1} + V_{2} be a free amalgam of the two vector spaces over V_{0}. Let ex_{i} : V_{i} → 𝔸(F_{i}), i = 0, 1, 2, denote ex_{|V_{i}}. We then have a natural homomorphism

ex′ : V_{1} + V_{2} → 𝔸(F_{1}) · 𝔸(F_{2}) ⊆ 𝔸(F),

defined by ex(v_{1} + v_{2}) = ex_{1}(v_{1}) · ex_{2}(v_{2}), for v_{1} ∈ V_{1}, v_{2} ∈ V_{2}.

By the theory of Abelian groups we have a direct decomposition

𝔸(F) = (𝔸(F_{1}) · 𝔸(F_{2})) × B.

Let

V = (V_{1} + V_{2}) × B

and ex : V → 𝔸(F) be defined as ⟨v, b⟩ ↦ ⟨ex′(v), b⟩. We then have that ker ex = ker ex_{0} ≅ ℤ^{N} and all the axioms of T_{A} satisfied for 𝕍 = (V, ex, 𝔸(F)) by construction. Thus 𝕍 is a model of T_{A} + {Λ ≅ ℤ^{N}} extending the amalgam 𝕍_{0} ⊆ 𝕍_{i}, i = 1.2□

Now, in the proof of Keisler’s theorem the unique model of cardinality ℵ_{1} is an Ehrenfeucht-Mostowski model, that is it realises countably many complete types over any countable set. Thus if we use the amalgamation property for countable models of Σ then a stronger version of Keisler’s Theorem holds In the presence of the amalgamation property Σ is ω-stable, that is the set of complete n-types over a countable model realisable in models of Σ is at most countable.

Corollary 6 T_{A} + {Λ ≅ ℤ^{N}} is ω-stable.

Under assumption of ω-stability and the amalgamation property for countable models Shelah develops the theory of splitting, ranks and independence.

Recall that a complete type p over B splits over C ⊆ B if there are b_{1}, b_{2} in B realising the same type over C and a formula ϕ(x, y) such that ϕ(x, b_{1}) ∈ p and ¬ϕ(x, b_{2}) ∈ p.

Non-splitting in ω-stable classes is an independence relation ⌣^{|} . More precisely for subsets A, B, C with C ⊆ A and C ⊆ B

A ⌣^{|}_{C} B iff tp(a/B) doesn’t split over C for all finite a in A.

In what follows we are interested in the case when A, B and C are models. It is helpful to notice that Lemma 4.3 In the theory of algebraically closed fields

F_{1} ⌣^{|}_{F_{0}} F_{2} iff the fields F_{1} and F_{2} are linearly disjoint over F_{0} = F_{1} ∩ F_{2}.

Proof Immediate from definitions. See [L], Chapter VIII, section 4 for the definition and properties of linear disjointness.□

By Fact 2 the elements and types of sort A in T_{A} + {Λ ≅ ℤ^{N}} are in a direct correspondence to types in the theory of algebraically closed fields. It follows

Lemma 4.4 Let 𝕍_{0} ⊆ 𝕍_{i}, i = 1, 2 be models of T_{A} + {Λ ≅ ℤ^{N}} intersecting at V_{0} and A_{i} = 𝔸(F_{i}) the sort A subset for 𝕍_{i}, and F_{i} the corresponding algebraically closed field, i = 0, 1, 2. Then

A_{1} ⌣^{|}_{A_{0}} A_{2} iff the fields F_{1} and F_{2} are linearly disjoint over F_{0} = F_{1} ∩ F_{2}.

By quantifier elimination we know that to check non-splitting of a general type of tp(x̄/V_{2}), x̄ in V_{1}, over V_{0} it is enough to check that corresponding algebraic varieties over A_{0} in (5) of Lemma 3.3 are irreducible over A_{2}, which follows if F_{1} and F_{2} are linearly disjoint over F_{0}. Thus we get

Corollary 7

𝕍_{1} ⌣^{|}_{𝕍_{0}} 𝕍_{2} iff the fields F_{1} and F_{2} are linearly disjoint over F_{0} = F_{1} ∩ F_{2}.

Finally observe by definitions

Remark Algebraically closed fields F_{1} and F_{2} are linearly disjoint over F_{0} = F_{1} ∩ F_{2} if and only if there is an algebraically independent set B such that

B = B_{1} ∪ B_{2}, B_{1} ∩ B_{2} = B_{0}

and B_{i} is a transcendence basis for F_{i} for i = 0, 1 and 2.

Now we can introduce the notion of excellency of a sentence Σ ( [Sh]). It is based on the notion of an independent n-system of countable models.

By this we mean a collection of countable models {M_{s} : s ⊂ n} of Σ (here n = {0, …, n−1} and ⊂ means the proper subset relation) with the property that

M_{s} ≺ M_{t} iff s ⊂ t

and

M_{s} ⌣^{|}_{M_{s∩t}} M_{t}, for any s, t ⊂ n.

Example and definition Let F be an algebraically closed field containing k_{0}, B ⊆ F a subset algebraically independent over k_{0} and

B = B_{1} ˙∪ … ˙∪B_{n}

its partition into non-empty subsets. Define, for s ⊂ n :

F_{s} = acl(k_{0}(⋃{B_{i} : i ∈ s})).  (13)

By the observation about linear disjointness {F_{s} : s ⊂ n} is an n-independent system of subfields. By the above remark characterising linear disjointness of algebraically closed fields any independent n-system of algebraically closed fields containing k_{0} has this form.

Proposition 4 An independent n-system of countable models of T_{A} + {Λ ≅ ℤ^{N}} has the form

{𝕍_{s} = (V_{s}, A_{s}) : s ⊂ n}, for A_{s} = 𝔸(F_{s})

where {F_{s} : s ⊂ n} is a system of the form (13).

Proof Immediate by Corollary 7 and following remarks.□

Definition A sentence Σ all of whose models are atomic is said to be excellent if the class of models of Σ has the amalgamation property for countable models and for any independent n-system of countable models {M_{s} ≺ M : s ⊂ n} there is a model M_{n} of Σ, such that M_{s} ≺ M_{n} for all s ⊂ n and M_{n} prime over {M_{s} ≺ M : s ⊂ n}.

Remark In particular, it follows that {M_{s} ≺ M : s ⊂ n} is good, that is any type over the set realisable in a model of Σ is principal.

Definition Given a countable algebraically independent over k_{0} subset B ⊆ F and its partition B = B_{1} ˙∪ … ˙∪B_{n} define acl-B-generated extension of k_{0} to be the field

k_{0}^{B} = k_{0}(⋃_{s⊂n} F_{s}), an extension of k_{0} by the algebraically closed fields.

Shelah’s Theorem [Sh] Suppose that all models of an L_{ω_{1},ω}-sentence Σ are atomic, Σ is categorical in ℵ_{n}, all n, and assume also that 2^{ℵ_{n+1}} > 2^{ℵ_{n}}, for all n ∈ ℕ (or just GCH). Then the class of models of Σ is excellent and ω-stable.

Assuming below GCH and that T_{A} + {Λ ≅ ℤ^{N}} is categorical in all uncountable powers we have

Corollary 8 For any algebraically independent countable B with a partition,

|S_{A}(𝕍(k_{0}^{B}))| = ℵ_{0}.

Proposition 5 Given 𝕍, a model of T_{A} + {Λ ≅ ℤ^{N}}, and an algebraically independent B ⊆ F (𝕍) with a partition, 𝕍 is L^{∗}-atomic over V (k_{0}^{B}).

Proof We can reduce any type over V (k_{0}^{B}) to the type of an n-tuple v = ⟨v_{1}, …, v_{m}⟩ ∈ V^{n} linearly independent over V (k_{0}^{B}). By Proposition 1(ii) and Corollary 8 any such type is L^{∗}-principal. □

5 Arithmetic consequences

Given an algebraically closed field F ⊇ k_{0} let, for a ∈ 𝔸(F),

a^{ℚ} = {x ∈ 𝔸(F) : x^{n} = a^{m} some m, n ∈ ℤ, n ≠ 0}.

In particular, 1^{ℚ} = 𝔸_{tors} is the set of all torsion points of 𝔸(F).

Given a subset X ⊆ 𝔸 we denote

k_{0}(X) = k_{0}(⋃{[x] : x ∈ X}) = dcl X ∩ F (𝔸).

In particular we consider for a given finite collection of points a_{1}, …, a_{n} ∈ A(F), n ≥ 0,

k_{a} = k_{0}(𝔸_{tors}, a_{1}^{ℚ}, …, a_{n}^{ℚ})

the subfield of F generated over k_{0} by all the coordinates of the elements of a_{i}^{ℚ}, including a_{0} = 1. In other words k_{a} is the field obtained by adjoining to k_{0} the coordinates of all the points in the division hull of the group generated by a_{1}, …, a_{n}. In particular, we need 𝔸_{tors} only in case n = 0.

We are going to consider the group 𝔸(k_{a}) of k_{a}-points of 𝔸, and the group 𝔸(k_{0}).

We denote K̃ = acl(K) for a field K.

Theorem 1 (Torsion points) Assuming ℵ_{1}-categoricity of T_{A} + {Λ ≅ ℤ^{N}} we have for any finitely generated extension k of k_{0} :

(i) the torsion subgroup 𝔸_{tors}(k) of 𝔸(k) is finite;

(ii) there is a number d such that for any l the Galois group Gal(k̃ : k) has at most d orbits on the set

{⟨a_{1}, …, a_{N}⟩ ∈ 𝔸_{l}^{N} : a_{1}, …, a_{N} generate 𝔸_{l}};

(iii) for all but finitely many prime p the group Gal(k̃ : k) acts on the Tate module T_{p}(𝔸) as GL_{N}(ℤ_{p}), and for remaining finite number of p the group acts as a subgroup of GL_{N}(ℤ_{p}) of finite index.

Proof In fact we are going to use only the fact that all models of infinite transcendence degree of the sentence are L^{∗}-atomic (Proposition 2). This property is preserved under expansion of the language with finitely many constant, thus we can assume that k = k_{0}.

(i) follows from (ii). Indeed, let a ∈ 𝔸(k_{0}) be a primitive solution of the equation x^{l} = 1. Then a can be included in some ā, a generating Ntuple of 𝔸_{l}. For any m co-prime with l we have that ā^{m} (component-wise) is generating 𝔸_{l} as well. But a and a^{m} are not conjugated by a Galois automorphism over k_{0}, unless m ≡ 1 mod l. Hence, by (ii), φ(l) ≤ d, so l is bounded.

(ii) Follows from Proposition 2 or, more precisely, from the fact that the type of any N-tuple τ pseudogenerating kernel stabilises. This implies that there are at most d complete types of pseudogenerating N-tuples, some finite d, and by ω-saturatedness of the structure on 𝔸 we get d orbits under the automorphism group, which acts on 𝔸(k̃_{0}) ⊇ 𝔸_{tors} as Gal(k̃_{0} : k_{0}).

(iii) Essentially the same argument as for (ii) but in a different language. Consider a model 𝕍 of T_{A} with Λ(V) ≅ ℤ̂^{N}, where ℤ̂ denotes the com-

pactification of ℤ in profinite topology. It is known that equivalently we can represent

ℤ̂ ≅ ∏_{p prime} ℤ_{p},  (14)

as a direct product of additive groups of p-adic numbers, considered as topological groups. We then have correspondingly

Λ(V) ≅ ∏_{p prime} ℤ_{p}^{N}  (15)

and each p-component of the direct product is a module over the ring ℤ_{p} which can be identified as the Tate module T (𝔸) of 𝔸.

Identifying Λ(V) with ℤ̂^{N}, notice that a pseudogenerating tuple τ of ℤ̂_{p}^{N} generates a dense subgroup of the group. Hence an α ∈ Aut(ℤ̂^{N}) is uniquely determined by the pseudogenerating tuple τ′ = α(τ). This α obviously preserves the profinite topology and is ℤ-linear so component-wise α is ℤ_{p}-linear, hence

Aut(ℤ̂^{N}) ≅ ∏_{p prime} GL_{N}(Z_{p}).

Any Galois automorphism α on 𝔸_{tors} over k_{0}, by elimination of quantifiers and Lemma 3.2, can be lifted to an automorphism of the additive group Λ(V). Since there are only finitely many types of generating tuples, the Galois automorphisms induce a subgroup Aut_{Gal}(ℤ̂^{N}) of Aut(ℤ̂^{N}) of finite index. Component-wise we have

Aut_{Gal}(ℤ̂^{N}) ≅ ∏_{p prime} G_{p}, G_{p} ⊆ GL_{N}(Z_{p})

with G_{p} = GL_{N}(Z_{p}) for almost all p and of finite index in the remaining finite number of p.□

Comments (i) of Theorem 1 and (i) of the next theorem for a number field k is an immediate consequence of the Mordell-Weil Theorem (and Dirichlet’s theory in the case 𝔸 is a one-dimensional torus). (iii) of Theorem 1 is well-known for the one-dimensional torus (the multiplicative group of the field) and is a difficult theorem of Serre for elliptic curves without complex multiplication. No other cases are known to the author.

The known cases of the next theorem are classically proved via the theory of Abelian Galois extensions and the method of infinite descent. For elliptic curves without multiplication (iii) is given by Bashmakov’s Theorem.

Theorem 2 (Kummer’s Theory and heights) Assuming ℵ_{1}-categoricity of T_{A} + {Λ ≅ ℤ^{N}} we have:

(i) 𝔸(k_{0}) ≅ A_{0} × 𝔸_{tors}(k_{0}) for some free abelian group A_{0};

and for a_{1}, …, a_{n} ∈ 𝔸 multiplicatively independent:

(ii)

𝔸(k_{a}) ≅ A_{a} × 𝔸_{tors} · a_{1}^{ℚ} · · · · · a_{n}^{ℚ}

for some A_{a} free abelian;

(iii) given b_{1}, …, b_{k} ∈ 𝔸(k_{0}) multiplicatively independent there is an l ∈ ℕ such that for any m ∈ ℕ

Gal(k_{0}(𝔸_{ml}, b1^{\frac{1}{ml}} , …, bk^{\frac{1}{ml}}) : k_{0}(𝔸_{ml}, b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}})) ≅ (ℤ/mℤ)^{Nk}.

(iv) given b_{1}, …, b_{k} ∈ 𝔸(k_{a}) such that a_{1}, …, a_{n}, b_{1}, …, b_{k} are multiplicatively independent, there is an l ∈ ℕ such that for any m ∈ ℕ

Gal(k_{a}(b1^{\frac{1}{ml}} , …, bk^{\frac{1}{ml}}) : k_{a}(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}})) ≅ (ℤ/mℤ)^{Nk}.

(v) Let F_{0} ⊆ F = F (𝔸) be a countable algebraically closed subfield and b_{1}, …, b_{k} ∈ 𝔸(F) multiplicatively independent over 𝔸(F_{0}). Then there is an l ∈ ℕ such that for any m ∈ ℕ

Gal(F_{0}(b1^{\frac{1}{ml}} , …, bk^{\frac{1}{ml}}) : F_{0}(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}})) ≅ (ℤ/mℤ)^{Nk}.

Proof We first prove (v). This follows directly from ω-stability and Proposition 1.

Now we consider (iv). We start with the remark that ω-stability and atomicity is preserved when the language is extended by naming a finite number of elements of a model. Thus if we name τ_{1}, …, τ_{N}, h_{1}, …, h_{n} ∈ V, such that τ_{1}, …, τ_{N} generate the kernel of a model and ex(h_{i}) = a_{i}, we still have atomicity of the model. Notice that such an expansion of a model 𝕍 names all elements of the subfield k_{a} = k_{0}(𝔸_{tors}, a_{1}^{ℚ}, …, a_{n}^{ℚ}) of F (𝕍). Hence we have that any sequence \{Wv^{\frac{1}{l}} : l ∈ ℕ\} associated with a v in V over k_{a} stabilises, by Proposition 1. Consider the sequence \{Wv^{\frac{1}{l}} : l ∈ ℕ\} associated with v_{1}, …, v_{k}, where ex(v_{i}) = b_{i}. We then have an l such that for any m all the k-tuples of roots of order m of ex(\frac{v}{l}) are conjugated by a Galois

automorphism over k_{a}(ex(\frac{v}{l})), that is over k_{a}(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}}). Then the group in (iv) is transitive on k-tuples of roots of order m of ex(\frac{v}{l}). It follows that the group action is of the form

ex(\frac{v}{ml}) ↦ α · ex(\frac{v}{ml}), for α ∈ 𝔸_{m}^{k}.

Hence the group in (iv) is isomorphic to 𝔸_{m}^{k}, and so to (ℤ/mℤ)^{Nk}.

The proof of (iii) is very similar, with the use of (i) instead of (ii) of Proposition 1. We thus get an l such that for any m all the (N + k)-tuples of roots of order m of ex(\frac{τ,v}{l}) are conjugated by a Galois automorphism over

k_{0}(ex(\frac{τ,v}{l})). In particular any two values of (b1^{\frac{1}{lm}} , …, bk^{\frac{1}{lm}}) are conjugated over

k_{0}(𝔸_{lm}, b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}}). Hence (iii) follows.

For (i) and (ii) we need to prove that the groups

A_{0} = 𝔸(k_{0})/𝔸_{tors}(k_{0}) and A_{a} ≅ 𝔸(k_{a})/𝔸_{tors} · a_{1}^{ℚ} · · · · · a_{n}^{ℚ}

are free.

Then (i) and (ii) follow by the general theory of Abelian groups, see [F, Th 14.4].

Obviously, A_{a} and A_{0} are torsion-free. By Pontryagin Theorem [F, Th 19.1] we only need to prove

Claim. For any finitely generated subgroup U ⊆ A_{a} (correspondingly U ⊆ A_{0}) the pure hull

Ũ = {u ∈ A_{a} : u^{m} ∈ U for some m ∈ ℕ}

of the subgroup in A_{a} (in A_{0}) is finitely generated.

Notice that U itself is free since it is finitely generated torsion-free [F, Th. 15.5].

Proof of Claim.

We prove it for the group 𝔸(k_{a}) using (iv) proved above and notice that the proof for 𝔸(k_{0}) is very similar but uses (iii) in place of (iv).

Let {u_{1}, …, u_{k}} be independent generators of U, and {b_{1}, …, b_{k}} elements in 𝔸(k_{a}) which correspond to {u_{1}, …, u_{k}} under the natural projection 𝔸(k_{a}) → A_{a}. Thus {a_{1}, …, a_{n}, b_{1}, …, b_{k}} are multiplicatively independent.

We claim that for some l

Ũ ⊆ gp(u1^{\frac{1}{l}} , …, uk^{\frac{1}{l}}).  (16)

(Here and below gp(S) for a subset S ⊆ A stands for a subgroup of A generated by elements of S.)

By (iv) we can choose an l ∈ ℕ such that for any m ∈ ℕ

Gal(k_{a}(b1^{\frac{1}{ml}} , …, bk^{\frac{1}{ml}}) : k_{a}(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}})) ≅ (ℤ/mℤ)^{Nk}.

If (16) does not hold for this l then there is g ∈ 𝔸(k_{a}) such that

g^{m} ∈ gp(𝔸_{tors}, a_{1}^{ℚ}, …, a_{n}^{ℚ}, b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}})

but g is not in the subgroup.

Then, replacing g by g · c, some c ∈ gp(𝔸_{tors}, a_{1}^{ℚ}, …, a_{n}^{ℚ}), we can have

g^{m} ∈ gp(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}}), g ∉ gp(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}}),

and

g ∈ gp(b1^{\frac{1}{lm}} , …, bk^{\frac{1}{lm}}).

The latter can be written as

b1^{\frac{s_{1}}{lm}} · · · · · bk^{\frac{s_{k}}{lm}} = g,  (17)

for some integers s_{1}, …, s_{k} ∈ {0, …, m − 1}. We may assume s_{1} ≠ 0. Then for α ∈ 𝔸_{m}, a torsion point of order m, by (iv)

(b1^{\frac{1}{lm}} , b2^{\frac{1}{lm}} , …, bk^{\frac{1}{lm}}) ↦ (αb1^{\frac{1}{lm}} , b2^{\frac{1}{lm}} , …, bk^{\frac{1}{lm}})

generates a Galois automorphism α̂ over k_{a}(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}}). By (17) α̂(g) = α^{s_{1}}g ≠ g, contradicting g ∈ 𝔸(k_{a}). Claim proved and the Proposition follows. □

The previous statements can be generalised to the fields of the form k_{0}^{B}, acl-B-generated extensions of k_{0}, introduced in section 4. Given finite subset a = {a_{1}, …, a_{r}} ⊆ F and k_{0}-algebraically independent B with an n-partition denote also

k_{a}^{B} = k_{0}^{B}(a_{1}^{ℚ}, …, a_{r}^{ℚ}).

Theorem 3 Assuming GCH and the categoricity of T_{A} + {Λ ≅ ℤ^{N}} in all uncountable cardinals, we have

(i)

𝔸(k_{a}^{B}) ≅ A_{a}^{B} × a_{1}^{ℚ} · · · · · a_{r}^{ℚ} · ∏_{s⊂n} 𝔸(F_{s}),

for some free abelian A_{a}^{B};

(ii) if b_{1}, …, b_{k} ∈ A are multiplicatively independent over ∏_{s⊂n} 𝔸(F_{s}) · a_{1}^{ℚ} · . . . · a_{r}^{ℚ}, there is an l ∈ ℕ such that for any m ∈ ℕ

Gal(k_{a}^{B}(b1^{\frac{1}{ml}} , …, bk^{\frac{1}{ml}}) : k_{a}^{B}(b1^{\frac{1}{l}} , …, bk^{\frac{1}{l}})) ≅ (ℤ/mℤ)^{Nk}.

Proof Obviously (ii) in the statement of the theorem is very similar to (iv) of Theorem 2 with k_{a} replaced by k_{a}^{B}. On the other hand by Corollary 8 and Proposition 5 we have ω-stability and L^{∗}-atomicity of models of the sentence over k_{0}^{B}, and this is the only fact we use to prove (iv) of Theorem 2. Thus (ii) follows by repeating the same argument. □

Remark This theorem has been proved without any assumptions for the one-dimensional torus 𝔸 (the multiplicative group of the field) in [Z0] in a stronger form: we don’t need B to be independent. Nevertheless the proof in [Z0] uses heavily the theory of linearly disjoint extensions of a field.

Now we want to show that the statements of Theorems 1 -3 imply excellency and ω-stability. Moreover, Theorem 4 Assume that the statements (ii) of Theorem 1, (iii) and (v) of Theorem 2 and (ii) of Theorem 3 hold. Then T_{A} +{Λ ≅ ℤ^{N}} is almost quasiminimal excellent in the sense of [Z3] and is categorical in all uncountable cardinalities.

Proof First we remark that (ii) of Theorem 1 implies, taking into account quantifier elimination and the description of types (Lemma 3.3), that any type over ∅ of a tuple of elements in the standard kernel is principal (in the language L^{∗}). (iii) of Theorem 2 in combination with the previous statement implies that any type over ∅ is principal.

Our proof of categoricity is based on the definition of almost quasiminimality and the categoricity theorem 5 in section 3 of [Z3].

Let B ⊆ 𝔸 be an irreducible algebraic curve in 𝔸 defined over a finite extension k of k_{0} and thus defined over some choice of constants in A. Notice that by the previous remark the type of constants is principal, hence if we prove categoricity of the class of models with the new constants we get also the categoricity of the original class. (In fact, Fact 2 implies that one can choose B defined over k_{0}).

Let U = LnB = {v ∈ V : ex(v) ∈ B}. By elimination of quantifiers B is strongly minimal and U is quasi-minimal. Moreover, since the algebraic closure acl (in model theoretic sense) of B contains F = F (A) and the algebraic closure of F contains 𝔸(F), we have A ⊆ acl(B). Obviously V = Ln(A) and so V ⊆ cl(U), where by definition

cl(X) = Ln(acl(exX)).

It is now easy to see that cl satisfies the Assumption 1 of [Z3].

The observation above stating that any type is principal proves ω-homogeneity over ∅. ω-homogeneity over models follows immediately from (v) of Theorem 2, by the same argument. This proves Assumption 2 of [Z3].

Finally, compairing definitions one sees that (ii) of Theorem 3 with r = 0 states in terms of [Z0] that the type of a tuple ⟨v_{1}, …, v_{k}⟩ in V such that ex(v_{i}) = b_{i} over a special set V (k_{0}^{B}) is principal, thus definable over a finite subset. This proves Assumption 3 and the theorem.□

Corollary 9 The conditions in Theorems 1 - 3 are equivalent to the statement that T_{A} + {Λ ≅ ℤ^{N}} is categorical in all uncountable cardinals.

References

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[L] S.Lang, Algebra, Addison-Wesley Publishing Co., Reading, Mass., 1965

[LB] H.Lange, Ch.Birkenhake, Complex Abelian Varieties, Springer-Verlag, Berlin Heidelberg, 1992

[M] D.Mumford, Abelian Varieties, Oxford University Press, 1970

[Sh] S.Shelah, Classification theory for non-elementary classes.I: the number of uncountable models of ψ ∈ L_{ω_{,}ω}(Q)., Parts A,B, Isr.J.Math., v.46, (1983), 212-240, 241-273.

[Z0] B.Zilber, Covers of the multiplicative group of an algebraically closed field of characteristic zero. Submitted to JLMS

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[Z2] ——–, Algebraic Geometry via Model Theory In: Contemp.Math. 131(1992) (part 3), 523-537

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Available on www.maths.ox.ac.uk/˜ zilber