Complex geometry and pseudo-analytic structures

Oberwolfach tutorial

Boris Zilber 19-23 July, 2004

I.Hrushovski construction of new structures ([H1],[H2] and [Z0])

Given a class of structures M with a dimension notion d we want to consider a new function f on M. Then on (M, f) we can calculate a predimension

δ(X) = d(X ∪ f(X)) − size(X).

Consider subclass of structures (M, f) which satisfy (Hrushovski inequality):

δ(X) ≥ 0 for any finite X ⊂ M.

Amalgamate all such structures to get a universal and homogeneous structure in the class. The resulting structure (M̃, f) will have a good dimension notion and a nice geometry.

Observation: If M is a field and we want f = ex to be a group homomorphism

ex(x_{1} + x_{2}) = ex(x_{1}) · ex(x_{2})

then the corresponding predimension must be

δ(X) = tr.deg(X ∪ ex(X)) − lin.dim(X) ≥ 0.

The Hrushovski inequality, in the case of complex numbers and ex = exp, is equivalent to

tr.deg(x_{1}, …, x_{n}, e^{x_{1}}, …, e^{x_{n}}) ≥ n

assuming that x_{1}, …, x_{n} are linearly independent. This is the Schanuel conjecture.

Pseudo-exponentiation

Consider the class of fields of characteristic 0 with a function ex:

K_{ex} = (K, +, ·, ex)

satisfying

EXP1: ex(x_{1} + x_{2}) = ex(x_{1}) · ex(x_{2}) EXP2: ker ex is a cyclic additive subgroup.

Consider the subclass satisfying the Schanuel-Hrushovski condition

δ(X) = tr.deg(X ∪ ex(X)) − lin.dim(X) ≥ 0.

Amalgamation process produces a universal and homegeneous (generic) structure of any given infinite cardinality K̃_{ex}(λ) : an algebraically closed field with pseudo-exponentiation.

More generally Pseudo-analytic functions F :

The scheme can be repeated with other (classical) analytic functions (or systems of functions) F provided we know

I. The functional equation for F.

II. The Generalised Schanuel condition for F (GSCH).

Questions.

1. Is there a canonical choice of K̃_{ex}(λ) for each λ > ℵ_{0}?

2. What is the stability status of K̃_{ex}?

3. Does

ℂ_{exp} ≅ K̃_{ex}(2^{ℵ_{0}}), or ℂ_{exp} ≡ K̃_{ex}

hold? Compare the two theories.

Two consequences of the construction of a generic K̃_{F} .

GSCH: The (generalised) Schanuel condition.

EC: For any free and normal (non-overdetermined) irreducible system of polynomial equations over A

P (x_{1}, …, x_{n}, y_{1}, …, y_{n}) = 0

there exists a generic solution satisfying

y_{i} = ex(x_{i}) i = 1, …, n.

Admissible transformations T:

x_{i}^{T} := m_{i,1}x_{1} + … + m_{i,n}x_{n}

y_{i}^{T} = y_{1}^{m_{i,1}} · . . . · y_{n}^{m_{i,n}}.

Free: P = 0 does not imply

x_{1}^{T} = const, nor y_{1}^{T} = const

under any admissible transforamtion Normal

tr.deg(x_{1}^{T} , …, x_{k}^{T} , y_{1}^{T} , …, y_{k}^{T} /{a, ex a}) ≥ k for any admissible transformation T of the generic solution (x, y) of P = 0.

Proof of EC. Given P over K construct a ’free’ extension K′ ≥ K with solutions for P (x, y) = 0 & y = ex(x).□

Other consequences of the construction: Pregeometry and closure operator: K̃ comes equipped with a dimension notion

d(X) = min{δ(XY) : Y ⊆_{fin}K̃} and the corresponding closure operator

y ∈ cl(X) ⇔ d(Xy) = d(X).

One can construct K̃ with the extra assumption of infinite dimensionality ID For each n there is X, d(X) ≥ n.

Lemma Suppose, for some class H_{F} , conditions GSCH, EC, and the class H_{F} are first-order axiomatisable. Then Th(H_{F})+ GSCH + EC axiomatises the generics of H_{F} . If also ID is axiomatisable, then the theory of every infinite dimensional K̃ is near model complete and κ-stable, provided the number of quantifier-free types over K̃ is at most card K + κ.

Corollary. The following instances of K̃ are ωstable, [Z7]:

Generic function (K, +, ·, f)

δ(X) = tr.deg(X ∪ f(X)) − size(X) ≥ 0.

Generic function with derivations (K, +, ·, f^{(i)} _{i∈ℤ})

δ_{i_{1},...,i_{k}}(X) =

= tr.deg(X∪f^{i_{1}}(X)∪⋯∪f^{i_{k}}(X))−k·size(X) ≥ 0,

δ(X) = min_{i_{1},...,i_{k}} δ_{i_{1},...,i_{k}}(X).

Raising to powers ([Z1],[Z2] and [Z2’]) analogue of (ℂ, +, ·, x^{r} _{r∈R}), x^{r} = exp(r ln x).

Formalisation: two-sorted structure K^{R} = (V, K), V is a R-vector space, K = (K, +, ·) a field of characteristic 0,

ex : V → K^{×} a group homomorphism.

GSCH: δ(X) =

= lin.dim_{R}(X)+tr.deg(ex X)−lin.dim_{Q}(X)+d ≥ 0,

(e.g. d = tr.deg R).

EC: For any free and normal system

L(x_{1}, …, x_{n}) = a & P (y_{1}, …, y_{n}) = 0

(L is R-linear, P a polynomial over ℚ(ex a)) there exists a solution satisfying

y_{i} = ex(x_{i}) i = 1, …, n.

Free: L = a does not imply x_{1}^{T} = const and P = 0 does not imply y_{1}^{T} = const under any admissible transforamtion Normal:

lin.dim_{R}(x_{1}^{T} , …, x_{k}^{T} /a)+tr.deg(y_{1}^{T} , …, y_{k}^{T} /ex a) ≥ k

for any admissible transformation T of the generic solutions x, y of L = a, P = 0.

Remark For any finitely generated subgroup Γ of (K^{×})^{n} there is a K^{R} with Γ definable. Hence the Lang property for K^{×} is expressible.

The Lang property for an semi-abelian variety A states: for any algebraic subvariety W of A there are finitely many semi-abelian subvarieties A_{i} ≤ A such that

Γ ∩ W = ∪_{i}(Γ ∩ A_{i}).

Proposition EC is first-order definable in K^{R}.

Proof. Use Ax’s theorem on Schanuel’s conjecture for function fields to get:

Claim A. Given W (e) ⊆ K^{n}, with W ⊆ K^{n+ℓ} algebraic subvariety defined over ℚ and e ∈ K^{ℓ}, there is a finite collection μ(W) = {M_{1}, …, M_{m}} of proper ℚ-linear subspaces of V^{n} such that for any ℚ-linear subspace N ⊆ V^{n}, b ∈ K^{ℓ}, and any positive dimensional atypical component S of the intersection W (e)∩ex(N)·b there is M ∈ μ(W) and s ∈ S such that S ⊆ex(M) · s.

Theorem 1[Z2’] EC, GSCH and ID are first-order definable in class K^{R}. The theory of a generic member K̃^{R} of the class is near model complete and superstable.

Proof. Use the Lang property, Claim A (Ax’s Thm) and an analysis similar to one in [Z2].

Theorem 1.1 Assume Schanuel’s conjecture. Let R ⊆ ℝ. Then the theory ℂ^{R} (V = ℂ, K = ℂ, ex = exp) is superstable and near model complete. In particular it follows from the theory that any normal free system of exponential sums equations with real powers has a solution in ℂ.

Proof. Uses the proof of Theorem 1 and Khovanski’s Theorem as in [Z1].

A.Wilkie has recently proved the following. Wilkie’s Theorem Let ℝ_{exp,sin_{|}} be the expansion of the real field by the exponentiation and the restricted sin and S ≺ ℝ_{exp,sin_{|}} its minimal elementary submodel. Let k_{1}, …, k_{n} ∈ ℝ + iℝ be complex numbers represented in ℝ_{exp,sin_{|}} such that k_{1}, …, k_{n} are independent over S in the sense of the pregeometry of o-minimal structure ℝ_{exp,sin_{|}}. Let R be the subfield of ℂ generated by k_{1}, …, k_{n}. Then the structure ℂ^{R} of raising to powers satisfies the inequality

lin.dim_{R}(X) + tr.deg(exp X) − lin.dim_{Q} ≥ 0.

Moreover, it satisfies the uniform version of Schanuel’s conjecture.

Theorem 1.2 Let R⊆ℝ and satisfy the assumptions of Wilkie’s Theorem. Then the theory ℂ^{R} is superstable and near model complete. Any normal free system of exponential sums equations with powers in R has a solution in ℂ. Proof. This is a special case of Theorem 1.1 combined with Wilkie’s Theorem.

Problems 1. Extend Ax’s Theorem on Schanuel’s conjecture for differential fields to GSCH for Weierstrass functions (see W.D.Brownawell and K.K.Kubota, Algebraic independence of Weierstrass functions, Acta Arithmetica, 33 (1977), 113-148).

2. Find the functional equation, GSCH and EC for the Weierstrass function 𝔭(ω, x) as a function of two variables.

The canonical generics. Non-elementary context.

We consider again the fields with pseudo-exponentiation K_{ex} = (K, +, ·, ex).

Remark EXP, GSCH, EC and ID are L_{ω_{1},ω}- definable.

Suggestion For the choice of a ’canonical’ K̃ of cardinality λ consider the one that is prime over a (pregeometry) basis B ⊆K̃, card B = λ.

Tasks (see [Z0])

1.Prove the existence and uniqueness of ’the canonical’ structure of cardinality λ.

2. Compare the canonical K̃_{ex} with ℂ_{exp}.

Observation. Any analytic structure on ℂ has the following property: If S is an analytic subset of ℂ^{n} of dimension 0 then card S ≤ ℵ_{0}. Define, for S ⊆K̃ definable over A

dim S = max{d(s/A) : s ∈ S}. Countable closure property CCP: If dim S = 0 then card S ≤ ℵ_{0}. Equivalently, cl(A) is countable, for finite A. Remark. CCP is L_{ω_{1},ω}(Q)-definable.

Theorem (Shelah). If the theory of K̃ in some class is ω-stable then the prime over a basis exists and is unique for each λ.

Moreover, the canonical model has CCP.

This is the case for the theory of a generic function.

Problem Is this the case in general?

Is the non-elementary theory ...+GSCH+EC+ID+CCP categorical in uncountable cardinals?

Theorem 2 (Uniqueness Theorem)

1. [Z4] EXP+GSCH+EC+ID+CCP is categorical in uncountable cardinals, so the canonical field with pseudo-exponentiation K_{ex} exists and is unique in each uncountable cardinal. Moreover, its L_{ω_{1},ω}- theory is ’nearly model complete’.

2. The canonical field with raising to powers K^{R} exists and is unique in each uncountable cardinal. Moreover, its L_{ω_{1},ω}-theory is ’nearly model complete’.

Problem Prove similar statements for pseudo-Weierstrass functions and the operation of raising to powers on elliptic curves.

Proof of the Uniqueness Theorem. Ingredients:

1. Non-elementary stability theory and the Categoricity Theorem for quasiminimal excellent classes.

2. The group cover of K^{×} is quasiminimal excellent. Arithmetic analysis.

3. K_{ex} is quasiminimal excellent.

Definition A class H is quasiminimal excellent if for every M ∈ H there is a closure operator cl

(i) For any X ⊆M the substructure cl(X) ∈ H. (ii) (M, cl) is a pregeometry.

(iii) M is ω-homogeneous over submodels.

Definition A subset C ⊆M will be called special if there is a cl-independent A⊆M and A_{1}, … A_{k} ⊆ A such that

C = ⋃_{i} cl(A_{i}).

(iv) Suppose C ⊆M is special and x ∈ cl(C). Then the type of x over C is defined over a finite subset C_{0} ⊆C.

Categoricity Theorem for quasiminimal excellent classes [Z8]. If H is quasiminimal excellent and has an infinite-dimensional model, then for every uncountable λ there is unique model of H satisfying CCP.

The group cover of K^{×} is a two-sorted structure (V, K) with V = (V, +), K = (K, +, ·) and

ex : V → K

a homomorphism onto K^{×}.

Theorem [Z3]. Each sort of group covers of K^{×} with standard kernel is quasiminimal excellent. (Hence the class is uncountably categorical.)

Proof. Uses Kummer’s Theory and the theory of heights for (iii) and the theory of linear disjointness for (iv).

Problem Prove similar statement for group covers of elliptic curves and abelian varieties.

Gavrilovich’s Theorem 1. For elliptic curves ω-stability (non-elementary) and ω-homogeneity over models holds.

2. For abelian varieties ω-stability holds.

The Liouville function

Definition (Wilkie) A function

f : ℂ → ℂ

given by

f(x) = ∑_{i≥1} c_{i}x^{i}

is called a Liouville function, if c_{i}^{−1} = a_{i} are integers satisfying

for every l ≥ 1, for all large i, |a_{i+1}| > |a_{i}|^{i^{l}}

Definition (Koiran and others) The theory of generic polynomial is a limit of theories ℂ_{p}, where

p(x) = ∑_{1≤i≤n} b_{i}x^{i}, n → ∞

are polynomial with generic coefficients.

Theorem. Let ℂ_{f} = (ℂ, +, ·, f) then

1.(Wilkie [W]) ℂ_{f} satisfies GSCH: For any distinct nonzero complex numbers z_{1}, …, z_{n}

tr.deg(z_{1}, …, z_{n}, f(z_{1}), …, f(z_{n})) ≥ n;

2.(Koiran [K]) ℂ_{f} satisfies EC and is a model of the (complete) theory ACF_{0}+GSCH+EC = the theory of generic polynomials;

3. ℂ_{f} can be characterized as the unique canonical (prime over a basis) model of ACF_{0}+GSCH+EC of cardinality 2^{ℵ_{0}}

(and the theory is ω-stable and near model complete).

III. Poizat’s fields and other exotics

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Definition A bad field is a structure (K, +, ·, G) with MR(K) = N > 1 and G < K^{×}, a multiplicative subgroup, MR(G) = 1.

Problem (197?) Do bad field exist?

Theorem ([BL] Baldwin and Holland, 2002) Yes, for each N > 1, if we drop the requirement that G is a group.

Theorem ([P2] B.Poizat, 2000) There exist an almost bad fields (K, +, ·, G) with MR(K) = ω × N and G < K^{×}, a multiplicative subgroup, MR(G) = ω.

Proof. Use Hrushovski construction.

Easy case, G = P is just a subset (of black points).

δ(X) = Ntr.deg(X) − size(X ∩ P).

Axioms: GSCH: For distinct x_{1}, …, x_{k} ∈ P,

Ntr.deg(x_{1}, …, x_{k}) − k ≥ 0.

EC: Let V ⊆ ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C) which has a point ⟨a_{1}, …, a_{k}⟩ ∈ V satisfying:

a_{i} ≠ a_{j} and a_{i} ∉ acl(C), i ≠ j, i ≤ k (P -free)

and

tr.deg(a_{i_{1}}, …, a_{i_{m}}) ≥ \frac{m}{N}, for any i_{1} < ⋯ < i_{m} ≤ k

(P -normal).

Then there is ⟨a_{1}, …, a_{k}⟩ ∈ V ∩ P^{k}.

Claim. GSCH and EC for (K, +, ·, P) are first order. Proof. Easy □ Difficult case, G is a subgroup (green points):

δ(X) = Ntr.deg(X) − mult.rk(X ∩ G).

Axioms: GSCH: For multiplicatively independent x_{1}, …, x_{k} ∈ G,

Ntr.deg(x_{1}, …, x_{k}) − k ≥ 0.

EC: Let V ⊆ℂ^{k} be an irreducible algebraic variety defined over a finitely generated subfield ℚ(C) and which is G-free and G-normal. Then there is ⟨a_{1}, …, a_{k}⟩ ∈ V ∩ G^{k}.

Claim. GSCH and EC for (K, +, ·, P) are first order. Proof. Uses Ax’s Theorem. □

Problem. Explain these examples analytically.

Solutions, for N = 2.

K = ℂ.

Case, P, subset of ℂ. Let ϵ, h be algebraic numbers, ϵ ∉ ℝ ∪ iℝ, h ∉ ℚ, hℝ ≠ ϵℝ. Let

P = {f(ϵt + hq) : t ∈ ℝ, q ∈ ℚ}

Theorem 3.1 [Z9] (ℂ, +, ·, P) is a model of Poizat’s “black points” theory.

Let

G = {exp(ϵt + hq) : t ∈ ℝ, q ∈ ℚ}

Theorem 3.2 [Z9]. Assume Schanuel’s conjecture. Then (ℂ, +, ·, G) is a model of Poizat’s “green points” theory.

Problem. Do the general case N. Remark Necessarily, the real dimension of the bad subset (P or G) must be \frac{2}{N} . Thus, the theory of fractional dimensions needs to be involved in the construction.

Let

G = {exp(ϵt + hm) : t ∈ ℝ, m ∈ ℤ}

Theorem 3.3 [Z6] Assume Schanuel’s conjecture. Then

1.Th(ℂ, +, ·, G) is superstable, U(ℂ) = ω · 2, U(G) = ω.

2. The Miller-Speissegger spiral G^{0} = exp(ϵℝ) is type-definable in (ℂ, +, ·, G) as the connected component of G.

3. The field of reals is L_{ω_{1},ω}-definable in (ℂ, +, ·, G).

Problems Do canonical models exist in class (K, +, ·, G)?

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Let ϵ = 1 + i, ϵ̄ = 1 − i, h ∈ ℝ ∖ ℚ

G = {exp(ϵt + 2πihm) : t ∈ ℝ, m ∈ ℤ},
G^{⊥} = {exp(ϵ̄t + 2πihm) : t ∈ ℝ, m ∈ ℤ}
Γ = exp(2πihℤ).

Theorem 3.4 [Z6]. Assume Schanuel’s conjecture. Then Th(ℂ, +, ·, G, G^{⊥}, Γ) is superstable. U(ℂ) = ω · 2, U(G) = ω, U(G^{⊥}) = ω, U(Γ) = 1.

Remark The ’torus’ ℂ^{×}/Γ with the induced structure has superstable theory and possesses a typedefinable involution J (an ’almost complex’ conjugation) corresponding to the involution

J : ϵt + ϵ̄s ↦ ϵt − ϵ̄s, t, s ∈ ℝ.

References

[Ax] J.Ax, em On Schanuel Conjectures, Annals of Mathematics, 93 (1971), 252 - 258

[BK] W.D.Brownawell and K.K.Kubota, Algebraic independence of Weierstrass functions, Acta Arithmetica, 33 (1977), 113-148

[BH] J.T.Baldwin and K.Holland Constructing ωstable structures: computing rank. Fund. Math. 170 (2001), no. 1-2, 1–20

[H1] E.Hrushovski, A New Strongly Minimal Set, Annals of Pure and Applied Logic, 62 (1993), 147-166

[H2]———, Strongly Minimal Expansions of Algebraically Closed Fields, Israel Journal of Mathematics 79 (1992), 129-151

[M] C.Miller Tameness in expansions of the real field , to appear in Logic Colloquium ’01”

[K] P.Koiran, The theory of Liouville functions, To appear in JSL

[P1] B.Poizat, Le carre de l’egalite, JSL (1999)

[P2] B.Poizat, L’egalite au cube, JSL (2000)

[W] A.Wilkie, Liouville functions, to appear in European Logic Colloquium Paris 2000

[S] 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]^{1} B.Zilber, Analytic and pseudo-analytic structures to appear in European Logic Colloquium Paris 2000

[Z1]———, Exponential sums equations and the Schanuel conjecture, J.London Math.Soc.(2) 65 (2002), 27-44

[Z2]———, Raising to powers in algebraically closed fields. JML, v.3, no.2, 2003, 217-238

[Z2’]———, Raising to powers revisited. Preliminary version 2004 on the web-page

[Z3] ——–, Covers of the multiplicative group of an algebraically closed field of characteristic zero. Submitted.

1Everything under [Z] is available on www.maths.ox.ac.uk/˜ zilber

[Z4] ——–, Pseudo-exponentiation on algebraically closed fields of characteristic zero, to appear in APAL

[Z5] ———, Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety, to appear in Ravello conference proceedings

[Z6]———-, A remarkable pseudo-analytic structure Preliminary version, 2004

[Z7]———-, Generic function with derivations, In: Logic and Algebra, (ed. Yi Zhang), pp. 85– 100, Contemporary Mathematics, Vol 302, AMS, 2002

[Z8]———-, A categoricity theorem for quasi-minimal excellent classes. to appear in Proc. of Ann-Arbor workshop, In Cont. Math. [Z9]———-, Bi-coloured fields on the complex numbers. to appear in JSL