A note on the model theory of the complex field with roots of unity

B.Zilber

February, 1990

We prove here a theorem on ω-stability of an algebraically closed field of characteristic zero with an extra unary predicate distiguishing a subgroup of roots of unity. The result was obtained during two months in 1990 which I spent in the USA, the first of the months at MSRI, Berkeley, and then at MIT. My visit was organized by Udi Hrushovski and I am greatly obliged to Udi for this and for the inspiration I have got by talking to him and other people during the two months. This particular theorem resulted from discussions with Greg Cherlin and Jim Loveys and the many wisdoms stocked in the libraries at MSRI and MIT and I am very grateful to all mentioned above.

I didn’t consider publishing the paper since it looked at that time an isolated result answering my own curiosity. But later Hrushovski unfolded now widely known powerful links between stability theory and diophantine geometry so the theorem became a meaningful piece of the picture. In another direction I used the method of the paper to establish properties of the model-theoretic torus in [2], section 5. Some people asked me on references to the theorem, that’s when I wrote this introduction and put the paper in public domain.

We use H.Mann’s result [1].

Definition Let c_{1}x_{1} + ⋯ + c_{n}x_{n} = 1 be a linear equation with integer coefficients. Call its solution ⟨u_{1}, …, u_{n}⟩ primitive if there is no equation with integer coefficients c′_{1}, …, c′_{n} with |c′| < |c|, compairing lexicographically, having the solution ⟨u_{1}, …, u_{n}⟩.

Theorem 1 (H.Mann) For any linear equation with integer coefficients there is no more than finitely many primitive solutions in complex roots of unity.

Lemma 0.1 Let M be an ω−stable structure, X ⊆ M infinite, U a new name of a unary predicate, corresponding to a subset U(M) of M. Assume ⟨M, U⟩ is ω−saturated. Then there is a superset X^{#} ⊇ X of the same power, such that for any tuple ȳ from U(M) rank(ȳ/X^{#}) = rank(ȳ/X^{#} ∩ U(M)) in the sense of M.

Proof Easy.□

Lemma 0.2 Let ℂ be the field of complex numbers, U the predicate on C for roots of unity, ⟨ℂ^{∗}, U^{∗}⟩ an ω−saturated extension of ⟨ℂ, U⟩. Let A be a maximal subset of U^{∗} such that (*) no u_{1}, …, u_{m} ∈ A satisfies a non-trivial multiplicative equation

x_{1}^{n_{1}} · . . . · x_{k}^{n_{k}} = 1.

Then A is algebraically independent in the sense of ℂ.

Proof Suppose towards a contradiction Σc_{m}x^{m} is a non-trivial polynomial with integer coefficients, m multiindices, having a solution ā in A. Choosing |c̄| minimal for the given ā we see that ā^{m_{1}}, …, ā^{m_{n}} (m_{i} the multiindices of the polynomial) is a primitive solution of the linear equation Σc_{m}x_{m}. Evidently, the condition on primitivity is definable, thus by Mann’s theorem ā^{m_{1}}, …, ā^{m_{n}} ∈ acl(0) in the sense of ⟨ℂ, U⟩. Hence ā^{m_{1}}, …, ā^{m_{n}} ∈ U. But this contradicts the definition of A.□

Lemma 0.3 Assume X = X^{#} in the sense of Lemma 0.1. Then any groupautomorphism of U^{∗} fixing X ∩ U^{∗} can be extended to an automorphism of ⟨ℂ^{∗}, U^{∗}⟩ fixing X.

Proof By model theory of abelian groups and the definition of X U^{∗} = U · D, U^{∗} ∩ X = U · D_{X}, (direct products) for D_{X} ⊆ D torsion-free with extraction of roots of any power from any element. Let A_{X} ⊆ A ⊆ D, A_{X} ⊆ D_{X}, both maximal satisfying (*) as in Lemma 0.2. Evidently D is a ”divisible envelope” of A and so is D_{X} relative to A_{X}. By Lemma 0.2 A is algebraically independent. By Lemma 0.1 A \ A_{X} is algebraically independent over X. Thus A can be extended to a subset A_{C} of the field ℂ^{∗} in such a way that A_{C} \ A_{X} is the transcendense base of ℂ^{∗} over X. Now any group-automorphism α : U^{∗} → U^{∗} which fixes X ∩ U^{∗} can be extended to ᾱ : ℂ^{∗} → ℂ^{∗} as follows: put B = α(A) B_{X} = A_{X} and define B_{C} ⊇ B so that B_{C} \ B_{X} is a transcendense base of ℂ^{∗} over X. Define ᾱ on X as identity, on A as α and on A_{C} \ A_{X} = A_{C} \ X as a bijection on B_{C} \ B_{X}. Further define ᾱ on whole of ℂ^{∗} as a field isomorphism using the fact that acl(X ∪ A_{C}) = ℂ^{∗} as well as acl(X ∪ B_{C}) = ℂ^{∗}. Since U^{∗} ⊆ dcl(X ∪ A) = dcl(X ∪ B), ᾱ coincides with α on U^{∗}.

Theorem 2 The cardinality of the set of complete types of the structure of the field of complex numbers with the predicate distinguishing the roots of unity over X is the same as that of X. Thus the theory is ω-stable.

Proof We assume any type over X is realized in ⟨ℂ^{∗}, U^{∗}⟩. Using the notation of the proof of Lemma 0.3 we recall that any element c of the structure is algebraic, thus its type over X ∪ A_{C} is determined by a one-variable formula ϕ(x̄, ā, v) where x̄, ā are the parameters from X and A_{C} \ X. As the proof of Lemma 0.3 shows the types of ā and ā′ over X coincide if the corresponding coordinates of the tuples are simultaniously from A \ X or A_{C} \ A. Hence the number of types over X realized in C^{∗} is |X|.

Generalizations. The same method is good for

1. U a subgroup of the group of roots of unity.

2. U_{1}...U_{n} subgroups generated by algebraically independent elements of the multiplicative group of any a.c.field K.

In both cases the field expanded by U_{i}’s is of the same degree of stability the groups U_{i} are. In the second case Mann’s theorem is not needed.

References

[1] H.Mann, On linear relations between roots of unity Mathematica, v.12(1965), 107-117

[2] B.Zilber, Quasi-Riemann Surfaces. In: Logic:From Foundations to Applications, European Logic Colloquium, ed. W.Hodges et al. Clarendon Press, 1996., 515-536