Quasi-Riemann surfaces

B.I. Zilber Kemerovo University Kemerovo, 650043, Russia e-mail: zilberkemucnit.kemerovo.su

The study of Zariski-type structures using a ”non-standard analysis ” approach [Z] showed that the infinitesimal neighborhoods in the structures carry rather rich structures of germs, resembling the complex analytic ones.

Starting from this point we introduce in the present paper structures with infinitesimal neighborhoods, each carrying a ring of ”holomorphic germs”. The definition follows the pattern of a compact Riemann surface and we call the structures quasi-Riemann surfaces. The basic field can be any algebraically closed field.

We develop a theory of analytic subsets of M^{n} and show that the analogy with usual complex analysis goes rather far, we reconstruct almost everything from the elementary theory of one complex variable and give a proof involving many-dimensional case. We conjecture that the basic theory of analytic subsets can be reconstructed in full, leading to a proof that quasi-Riemann structures are Zariski-type.

We discuss several examples and a problem of identifying a quasi-Riemann surface as an algebraic curve. An important remark here is that though the problem is algebro-geometric in form a solution hardly can be obtained without a deep model-theoretical analysis.

Notice that one can define analogues of many-dimensional complex manifolds in the similar way, though the analytic theory will become more complicated. We have not done it for the sake of simplicity.

The paper was finished during my stay at the University of Helsinki and I would like to express my sincere thanks to Helsinki Logic Group for their hospitality and interest in my work.

I would like to thank the referee of the paper who pointed out several errors and many misprints and forced me to revise the paper.

1 C-structures and their specializations

Definition Let M be a first order structure in a relational language L and let C be a distinguished subcollection of the definable (with parameters) subsets of M^{n}, n = 1, 2, . . .. ⟨M, C⟩, or M, is a C-structure if it satisfies (C): universum of the structure is in C; the graph of equality is in C; any singleton of the universum is in C; disjunctions and conjunctions of closed relation are in C; cartesian products of closed relations are in C.

It follows from the axioms that for any n C restricted to M^{n} forms a base of closed sets of a topology which will be called C-topology (on M^{n}). Below ”closed” means closed in this topology. The language L is said to be C-language if all its basic relations are closed.

Below all structures are in language L and considered C-structures for some C.

Example 1

(i) Let M = k be a field. The basic relations of L are x + y = z and x · y = z. C consists of all positive Boolean combinations of the sets of solutions in k of polynomial equations, i.e. of the Zariski closed subsets of k^{n}. (ii) Let M = P(k) the projective line over the field k. More precisely, we choose the representation P(k) = k ∪ {∞} of the projective line. The basic relations of L are the relations distinguishing the Zariski closed subsets of P(k)^{n} as defined in any textbook on Algebraic Geometry. We can briefly explain here this notion as follows. The representation above gives the natural embeddins of k into P(k) as well as another embedding ^{−1} : x → x^{−1} of k, where 0^{−1} = ∞. Then P(k) = k ∪ k^{−1}, where k^{−1} is the image under ^{−1}, is the canonical affine cover of P(k) and it induces the canonical affine cover of P(k)^{n} for any n and also introduces the affine coordinates x_{i} and x_{j}^{−1} on the affine charts.

Then S ⊆ P(k)^{n} is said to be Zariski closed if its intersection with any of the affine charts is the zero-set of a system of polynomial equations in affine coordinates. Again C is the family of all Zariski closed sets. k and k^{−1} are open subsets of P(k). Notice that if k^{∗} ≻ k then P(k)^{∗} = P(k^{∗}) ≻ P(k) in the corresponding languages.

Let M^{∗} ≻ M be an elementary extension of M, S a definable relation or subset of M^{n}. In the paper S^{∗} or S(M^{∗}) stand for the set of realizations of the relation S in M^{∗}. Notice that for S ⊆ M^{k+n} closed and a ∈ M^{k} the set S ∩ {a} × M^{n} is closed by the axioms. If in addition, as is assumed below, the images of closed sets under projections pr : M^{k+n} → M^{n} are closed, then the set

S(a, M^{n}) = {b ∈ M^{n} : ⟨a, b⟩ ∈ S} = pr(S ∩ {a} × M^{n})

is closed too.

Definition Let M^{∗} ≻ M be an elementary extension of M, A ⊆ M^{∗}. A mapping π : A → M will be called a (partial) specialization, if π(S^{∗} ∩ A^{n}) = S ∩ π(A^{n}) for any n-ary closed S.

Remark By the definition a specialization is an identity on M, since any singleton {s} is closed.

Definition A C-structure M will be called quasi-compact if it satisfies: (P) the images pr(S) of definable closed subsets under the projections pr : M^{k+n} → M^{n} are closed. and (QC) For any finitely consistent family {S_{t} : t ∈ T } of definable closed subsets of M^{n}

⋂_{t∈T} S_{t} is non-empty.

Proposition 1 Suppose M is a quasi-compact structure, M^{∗} ≻ M. Then there is a specialization π : M^{∗} → M, moreover, any partial specialization can be extended to a total one.

Proof Suppose π is defined on some D ⊆ M^{∗} and b′ an element of M^{∗}. We want to extend the definition of π to D ∪ {b′}. To do this consider

{S(x, d̄′) : S is a closed relation and S(b′, d̄′), d̄′ ∈ D^{n}}.

Now look at the corresponding family

{S(M, d̄) : d̄= π(d̄′), d̄′ ∈ D^{n}}.

This is a finitely consistent family, by (P), of closed sets. Thus by (QC) the intersection of the sets of the family is non-empty. Put π(b′) = b for a b chosen in the intersection. It is immediate that the extended map is again a partial specialization. Continuing this process one gets a total specialization. □

Definition Let M be a structure in L. Suppose M^{∗} ≻ M, and π : M^{∗} → M is a map identical on M. We will call a definable set (relation) A ⊆ M^{n} π-closed if π(A^{∗}) ⊆ A. (Evidently, ⊆ can be replaced here by =.) The family of all π-closed relations is denoted C_{π}. Evidently C_{π} satisfies (C), thus M becomes C_{π}-structure.

Remark The above definition is applicable also to type-definable subsets A of M^{n}, i.e. A = ⋂{A_{i} : i ∈ I} for a family A_{i} of definable subsets. It follows easily from the definition that if {A_{i} : i ∈ I} consists of π-closed sets then A is π-closed. Thus π-closedness is a topological notion.

Proposition 2 For any M^{∗} and π as in the definition above the C_{π}-structure M is quasi-compact, provided M^{∗} is saturated over M (i.e. any n-type with parameters in M is realized in M^{∗}).

Proof It is immediate from the definition that (P) holds for M. To check (QC) consider any finitely consistent family {C_{t} : t ∈ T } of closed subsets of M^{n}. Then ⋂{C_{t}^{∗} : t ∈ T } ≠ ∅, since M^{∗} is saturated, and thus contains a point c′. Then π(c′) ∈ ⋂{C_{t} : t ∈ T } by the closedness of C_{t} and (QC) is proved. □

Definition Given M^{∗} ≻ M and π we call an infinitesimal neighborhood of a ∈ M^{n} the subset in M^{∗} given as

V_{a} = π^{−1}(a).

Clearly then for a, b ∈ M we have V_{(a,b)} = V_{a} × V_{b}.

Lemma 1.1 A definable subset A ⊆ M^{n} is C_{π}-closed iff for every a ∈ M^{n} V_{a} ∩ A^{∗} ≠ ∅ implies a ∈ A; correspondingly, A is C_{π}-open iff for every a ∈ A V_{a} ⊆ A^{∗}. Also Int(A) = {a ∈ A : V_{a} ⊆ A^{∗}}.

Proof Immediately from the definitions. □

Proposition 3 Let K be an algebraically closed field, K^{∗} ≻ K be saturated over K, π : P(K^{∗}) → P(K) a specialization respecting Zariski closed relations. Then P(K) is C_{π}-topologically connected.

Proof Consider E = E(K) ⊆ P(K) × P(K) a (closed) elliptic curve over K which, as is well known, is equipped with a divisible algebraic group structure. Since the projection E → P(K) covers P(K) it suffies to show that E is connected. Suppose the contrary, E = A ˙∪B, A ≠ ∅, B ≠ ∅ both being closed and open in E. By the definition of the topology and (QC) both A and B are definable. Then the subgroup

H = {h ∈ E : h + A = A} = {h ∈ E : h + B = B}

is definable and proper. Consider H^{∗}, A^{∗} and B^{∗}. Since A is open, V_{a} ∩E^{∗} ⊆ A^{∗} if a ∈ A. Also, since + is continuous, V_{0} ∩ E^{∗} + V_{a} ∩ E^{∗} = V_{a} ∩ E^{∗}. It follows H^{∗} ⊇ V_{0} ∩ E^{∗} and for all h ∈ H

V_{h} ∩ E^{∗} = h + V_{0} ∩ E^{∗} ⊆ H^{∗}

which means H is open in E. It follows from (QC) that the cover

E = ⋃_{c∈E} c + H

is finite, hence H is of finite index in E which contradicts the divisibility. □

Lemma 1.2 Let M be a structure in L equipped with a compact Hausdorff topology τ which induces the product topology (also called τ) on M^{n} for any n and all τ-closed sets are definable, M^{∗} ≻ M is saturated over M. Then there is unique specialization π : M^{∗} → M respecting τ, i.e. π(S^{∗}) = S for any τ-closed set S ⊆ M. Moreover, C_{π}-topology coincides with τ.

Proof The existence is given by Proposition 1. π is unique since for any a′ ∈ M^{∗} π(a′) = a iff a′ ∈ U^{∗} for any open neighborhood U of a. This also means that

V_{a} = ⋂_{a∈U} U^{∗}.

Suppose now that A is π-closed. Then, by the saturatedness of M^{∗}, a is in τ-closure of A iff V_{a} intersects A^{∗} and this is equivalent by Lemma 1.1 to a ∈ A. Hence A is τ-closed. □

Definition The specialization given by the Lemma above is said to be the standard part mapping.

Example 2 Let k be a locally compact field. This, as follows from the general classification [Bo,VI, 9], implies that the topology is given by a valuation | | : k → R_{>0}. It follows from the local compactness that the projective line P(k) is compact in the topology of k. Let K ⊃ k be an algebraically closed field. It is known that | |, and thus the topology, can be extended onto K. It follows from definitions that any Zariski closed relation S on P(K) is closed in the topology. Thus the relativization of S to P(k) is closed in P(k). Let k^{∗} ⊆ K^{∗} be such that k^{∗} ≻ k and K^{∗} ≻ K in the language of fields with the valuations. Consider the standard part mapping π_{k} : P(k^{∗}) → P(k) for the Hausdorff topology given by the valuation. As was noticed above π_{k} can be viewed as a partial specialization P(K^{∗}) → P(K) respecting Zariski closed relations. By Proposition 1 there is a specialization π ⊇ π_{k}, π : P(K^{∗}) → P(K) respecting Zariski closed relations and the unary predicate distinguishing P(k). We thus came to C_{π}-topology on P(K) which coincides with the compact Hausdorff topology on P(k) and is quasi-compact. If K ≠ C then this topology is different from one given by the non-Archimedean valuation on K since the latter can not be quasi-compact for an algebraically closed K.

2 Existentially closed specializations

Definition For a specialization π : M^{∗} → M we say that the pair (M^{∗}, π) is λ-e.c. if for any M′ ≻ M^{∗} ≻ M, any finite ā ⊆ M′, A ⊂ M^{∗} of cardinality less than λ and a specialization π′ : ā ∪ M^{∗} → M, π′ ⊇ π, there is an elementary isomorphism over M ∪ A α : ā → M^{∗}, such that π′ = π ∘ α on ā. We omit mentioning λ if λ = ω.

Proposition 4 Let k be a valued locally compact field, k^{∗} ≻ k in the language containing all closed (in the valuation topology) relations, and k^{∗} is λ-saturated for λ > card(k). Let π : P(k^{∗}) → P(k) be the standard part mapping. Then (P(k^{∗}), π) is λ-e.c.

Proof Let k′ ≻ k^{∗}, π′ : P(k′) → P(k) and π′ extends π. Let for a′ ∈ k′ p(x) be the type of a′ over k ∪ A for some A ⊆ k of power less than λ. W.l.o.g. we assume that π′(a′) = a ≠ ∞. Let

F_{a} = \{|x − a| ≤ \frac{1}{n} : n ∈ N\}.

We claim that F_{a} ⊆ p(x) for otherwise |a′ − a| ≥ \frac{1}{n} for some n and then |π′(a′) − a| ≥ \frac{1}{n}. Take now b to be any element of k^{∗} satisfying p and we get π(b) = a since F_{a} ⊆ p(x). □

Proposition 5 For any quasi-compact structure M and cardinality λ there is an λ-e.c.pair (M^{∗}, π).

Proof We construct M^{∗} inside a monster model and π by the following routine process: Start with any M_{0} ≻ M and a specialization π_{0} : M_{0} → M. We will construct a chain of length λ of elementary extensions M_{i} ≻ M_{i−1} ≻ … ≻ M_{0} and specializations π_{i} : M_{i} → M, π_{i} ⊇ π_{i−1}. To construct M_{i+1} and π_{i+1} we first enumerate all triples of the form ⟨A, ā, p(x̄)⟩, where A is a λ-subset of M_{i}, ā ∈ M^{n} and p(x̄) is an n-type over M ∪ A. Let κ be the cardinality of the set of the triples. Now we construct specializations π_{i,j}, j ∈ κ, such that π_{i,j} ⊃ π_{i}, and their domains N_{i,j} : N_{i,0} = M_{i} and π_{i,0} = π_{i}. N_{i,j+1} = N_{i,j} ∪ {b̄}, where b̄ realizes p_{j}, if there is a specialization π′ ⊃ π_{i,j}, sending b̄ to ā_{j}. In this case π_{i,j+1} is π′. Otherwise, N_{i,j} = N_{i,j+1}, π_{i,j+1} = π_{i,j}. On the limit steps take unions. Put now M_{i+1} to be any model containing N_{i,κ}, and π_{i+1} ⊃ π_{i,κ}, a specialization from M_{i+1} to M. It follows from the construction, that for any M′ ≻ M_{i} ≻ M, any finite ā ⊆ M′, A ⊂ M_{i} of cardinality less than λ and a specialization π′ : ā ∪ M_{i} → M extending π_{i} there is an elementary isomorphism over M ∪ A α : ā → M_{i+1}, such that π′ = π_{i+1} ∘ α on ā. Now take M^{∗} = ⋃_{i<μ} M_{i}, where cf(μ) ≥ λ, π = ⋃_{i<μ} π_{i}. □

Corollary 1 Let M_{0} ≻ M be saturared over M and π_{0} : M_{0} → M a specialization. Then for any λ there is M^{∗} ≻ M_{0} ≻ M and a specialization π : M^{∗} → M extending π_{0} such that (M^{∗}, π) is λ-e.c. for C_{π}-structure M.

Lemma 2.1 Suppose, given an e.c. pair (M^{∗}, π) for M, a′ ∈ V_{a}, D(x, y) is an M-definable relation. Then there is b′ ∈ V_{b} ∩D(a′, M^{∗}) iff the following type is consistent:

p(y) = {D(a′, y)} ∪ {¬Q(c′, y) :⊧ ¬Q(c, b), Q closed , c′ ∈ M^{∗}^{m}, π(c′) = c}

Moreover, if the type is consistent, then for any b′ realizing p in some extension and any finite A ⊂ M^{∗} there is b′′ ∈ V_{b} ∩ D(a′, M^{∗}), such that tp(b′′, A ∪ M) = tp(b′, A ∪ M).

Proof It follows from the definition of specialization that p is consistent if such b′ exists. For the converse just verify that if b′′ realizes p, then an extension π′ ⊇ π to b′′, defined as π′(b′′) = b, is a specialization from M^{∗}∪{b′′} to M. The rest follows from the fact that (M^{∗}, π) is an e.c. pair.□

3 Quasi-Riemann surfaces

We start here with some e.c. pair ⟨M^{∗}, π_{M}⟩ for a quasi-compact C_{π_{M}} -structure M, K an algebraically closed field and some e.c. specialization π_{K} : P (K^{∗}) → P (K) preserving Zariski relations. Below we use the same notation π both for π_{M} and π_{K} as well as for the corresponding notions of infinitesimal neighborhoods, provided there is a way of understanding which of them we have in mind.

Definition A ring O_{b} of holomorphic germs in b ∈ M^{n} is a set of functions ϕ : V_{b} → K^{∗} closed under + and · (the operations on K^{∗}), such that: (1)

(i) ϕ(b) ∈ K; (ii) ϕ(V_{b}) ⊆ V_{ϕ(b)}; (iii) any constant function c : V_{b} → K is in O_{b};

(2)

(i) O_{b} is a local regular ring with the maximal ideal M_{b} consisting of those germs of O_{b} which have value 0 in b; (ii) dimension of O_{b} is n, which means that the dimension of vector space M_{b}/M_{b}^{2} over K is n; any set x_{1}, …, x_{n} of germs belonging to M_{b} which forms a basis of vector space M_{b}/M_{b}^{2} over K is said to be a system of parameters at b; (iii) O_{b} satisfies the Weierstrass Theorems: regularity: For any non-zero h ∈ M_{b} there is a K-linear nondegenerate transformation of parameters

⟨x_{1}, … x_{n}⟩ ↦ ⟨y_{1}, … y_{n}⟩

such that h is regular for y_{1} with respect to ⟨y_{1}, … y_{n}⟩, which means there is a point in V_{b} where y_{2} = … = y_{n} = 0 and h ≠ 0; preparation: For any h ∈ M_{b} regular for x_{1} with respect to ⟨x_{1}, … x_{n}⟩ there are germs w , g such that h = w · g, g is invertible in O_{b} and w is the Weierstrass polynomial, i.e one of the form x_{1}^{k} + a_{1}x_{1}^{k−1} + … + a_{k}, where a_{i} ∈ M_{b} and a_{i} ∈ O_{b}(x_{2}, …, x_{n}) and the latter is the subring of those germs ψ ∈ O_{b} which do not depend on x_{1}, i.e. the value of ψ is determined uniquely once the values of x_{2}, …, x_{n} are given; (3) for any system x_{1}, …, x_{n} of parameters the mapping

⟨x_{1}, …, x_{n}⟩ : V_{b} → V_{0}^{n}

is a bijection.

Remark In fact the definition says O_{b} satisfies essential abstract algebraic properties the classic local rings of germs of holomorphic functions do. See [GR, ch.II].

The following are standard algebraic consequences of the definition (cf. [Bo, ch.VII]).

Proposition 6 (Division Theorem) Given h ∈ O_{b} and a Weierstrass polynomial w for x_{1} there are unique g ∈ O_{b} and r ∈ O(x_{2}, …, x_{n})[x_{1}] such that

h = w · g + r

and degree of r with respect to x_{1} is less than that of w.

Proposition 7 O^{n} is a unique factorization domain.

Proposition 8 O^{n} is noetherian.

Proposition 9 For any ϕ ∈ O_{b} there is a unique sequence {Φ_{i} : i ∈ N} of homogeneous polynomials of degree i of x_{1}, …, x_{n} such that

ϕ − (Φ_{0} + Φ_{1} + ... + Φ_{m}) ∈ M_{b}^{m+1}

(the Taylor series of ϕ) and ϕ ∈ M_{b}^{m} iff Φ_{i} = 0 for all i < m.

Definition A quasi-Riemann surface (qRs in short) over a π-field K is defined by a quasi-compact two-sorted C_{π}-structure ⟨M, P (K)⟩ and its elementary extension ⟨M^{∗}, P (K^{∗})⟩ which forms an e.c. pair with π. M is assumed connected in C_{π}-topology and for any point b ∈ M^{n} there is given a ring O_{b} of dimension n of holomorphic germs and the rings O_{b} agree, i.e.

(i) for any b_{1} ∈ M^{k_{1}}, b_{2} ∈ M^{k_{2}}, ⟨b_{1}, b_{2}⟩ = b ∈ M^{k_{1}+k_{2}} there are embeddings

i_{1} : O_{b_{1}} → O_{b}, i_{2} : O_{b_{2}} → O_{b}

given as

i_{1} : ϕ(v_{1}) ↦ ϕ(v_{1}, v_{2}), i_{2} : ϕ(v_{2}) ↦ ϕ(v_{1}, v_{2})

such that any system of parameters x_{1,1}, …, x_{1,k_{1}} of the ring O_{b_{1}} jointly with paremeters x_{2,1}, …, x_{2,k_{2}} of O_{b_{2}} form a system of parameters of the ring O_{b};

(ii) for any ϕ ∈ O_{b} the mappings

ϕ(b_{1}, v_{2}) : V_{b_{2}} → K^{∗}, ϕ(v_{1}, b_{2}) : V_{b_{1}} → K^{∗}

are germs from O_{b_{2}}, O_{b_{1}}, correspondingly.

Remark Given a qRs M over a π-field K one necessarily gets also P(K) to be a qRs over K since the bijections x + c : V_{b} → V_{c} (c ∈ K) and x^{−1} : V_{b} → V_{∞}, given by a parameter x ∈ O_{b} identifies the infinitesimal neighborhoods of M and P(K) and transfers the structures of the rings of germs.

Lemma 3.1 For any non-constant germ ϕ at point a ∈ M^{n} of a qRs M

ϕ(V_{a}) = V_{ϕ(a)}.

Proof By the remark above we assume M = P(K). Also assume ϕ(a) = 0. Let x_{1}, … x_{n} be a system of parameters at a such that ϕ is regular for x_{1}. Let y be a parameter at 0. Then x_{1}, … x_{n}, y is a system of parameters at ⟨a, 0⟩ such that germ ϕ − y is regular for x_{1}. Hence

ϕ − y = (x_{1}^{k} + a_{1}x_{1}^{k−1} + … + a_{k}) · g

for some k, a_{i} ∈ M_{⟨a,0⟩} ∩ O_{⟨a,0⟩}(x_{2}, …, x_{n}, y) and invertible germ g at ⟨a, 0⟩. Take now any v ∈ V_{0}. We want to find w ∈ V_{a} such that ϕ(w) = v. Let α_{1}, … α_{k} be the values of a_{1}, … a_{k} corresponding to y = v, x_{2} = … = x_{n} = 0. Find a solution v_{1} of the equation for x_{1}

x_{1}^{k} + α_{1}x_{1}^{k−1} + … + α_{k} = 0.

v_{1} ∈ V_{0} since π(α_{1}) = … π(α_{k}) = 0. Hence we can take w to be the point in V_{a} where x_{1} = v_{1}, x_{2} = 0, … x_{n} = 0. □

Definition Let M be a qRs, U ⊆ M^{n} be open, a definable function f : U → K is said to be holomorphic (analytic) on U if

f^{∗}|V_{b} ∈ O_{b} for any b ∈ U;

a definable function f : U → P(K) is said to be meromorphic on U if

f^{∗}|V_{b} ∈ O_{b} or f^{∗}(V_{b}) ⊆ V_{∞} and \frac{1}{f^{∗}|V_{b}} ∈ O_{b} for any b ∈ U (here \frac{1}{f^{∗}(b)} = 0);

a definable subset S ⊆ U is said to be analytic in U if for any b ∈ U there are germs g_{1}, …, g_{k} ∈ O_{b} (the set of defining germs) such that

S^{∗} ∩ V_{b} = {z ∈ V_{b} : g_{1}(z) = … = g_{k}(z) = 0}.

Remark In the definition one may allow the set of defining germs to be infinite. This is equivalent by noetherianity.

Example 3 For M = P(K) in the language of Example 1, (M^{∗}, π) existentially closed, for each a ∈ P(K)^{n} one can take O_{a} to be the ring of germs of rational functions: i.e. of the form \frac{f}{g} with f and g polynomials in variables v_{1}, … v_{n}, g(a) ≠ 0. In case the i-th coordinate of a is ∞ replace v_{i} by v_{i}^{−1} in the definition. Then, by e.c., \frac{f_{1}}{g_{1}} and \frac{f_{2}}{g_{2}} represent the same germ, i.e. coincide on V_{a}, iff the two rational functions coincide on a Zariski open neighborhood of a. The isomorphism type of the rings are equal for all a ∈ P(K). This ring satisfies the definitions but evidently is too small to give an interesting analytic structure on P(K).

Example 4 Fix a field K and an ordered abelian group Γ written additively. The power series field K^{Γ} consists of all formal series

x = ∑_{γ∈Γ} a_{γ} · t^{γ}

with “coefficients” a_{γ} ∈ K, such that the support of x, supp(x) = {γ ∈ Γ : a_{γ} ≠ 0}, is a well ordered subset of Γ. These series are added and multiplied in the usual way and form a field (see [F], [R]). We consider K as a subfield of K^{Γ} by identifying c ∈ K with c · 1.

Note that π : K^{Γ} → P(K) given by

π(x) = \{ \begin{matrix}0 & if min(supp(x)) > 0 \\ ∞ & if min(supp(x)) < 0 \\ a_{0} & otherwise \end{matrix}

is a field-specialization of K^{Γ}. Note also that Vā = π^{−1}(ā) for ā = ⟨a_{1}, … a_{n}⟩ ∈ P(K)^{n} is just V_{a_{1}} ×…×V_{a_{n}}, V_{a} = a + V_{0}, if a ≠ ∞, V_{∞} = V_{0}^{−1} and V_{0} = the power series with positive powers only = V. In general K^{Γ} is algebraically closed, provided K is algebraically closed and Γ divisible. The proof in fact is the same as for fractional power series field in [W]. Let X = (X_{1}, …, X_{n}) and let K[[X]] denote the ring of formal power series ∑ c_{i}X^{i}, where i = (i_{1}, …, i_{n}) ranges over N^{n}, c_{i} ∈ K and X^{i} = X_{1}^{i_{1}} ⋯ X_{n}^{i_{n}}. It is well known (see [Bo, ch.VII, 3]) that the formal power series rings satisfy the assumptions (2) for the rings of germs. If v̄ = ⟨v_{1}, …, v_{n}) ∈ V^{n} and f = ∑ c_{i}X^{i} ∈ K[[X]], then f(v̄) = ∑ c_{i}v^{i} is a well defined element of K^{Γ} since only finitely many terms c_{i}v^{i} contribute a nonzero coefficient to a given monomial t^{γ} and the union of the supports of the elements c_{i}v^{i} is well ordered. (The case n = 1 is treated in [F], and the general case is similar.) One can see easily that for any f ∈ K[[X]] the map v̄ ↦ f(v̄) sends V^{n} into V_{f(0)} ⊆ K^{Γ}.

The maps v̄ ↦ f(v̄) can be considered on any Vā, ā ∈ P(K)^{n}, by shifting the domain to V^{n}, denote the maps f(Xā). More precisely, the shift X ↦ Xā is defined coordinatewise as X_{i} ↦ X_{i}−a_{i} if a_{i} ≠ ∞, and X_{i} ↦ X_{i}^{−1} if otherwise.

Define O^{n} attached to Vā to be K[[Xā]]. By what was discussed above one sees that O^{n} satisfies the assumptions for rings of germs. Thus :

Let K be algebraically closed, Γ divisible and L a language for P(K) and P(K^{Γ}) such that we have P(K^{Γ}) ≻ P(K), the extension is saturated over P(K) and π_{Γ} is e.c. Then M = P(K) is qRs over K.

Problem Giving a qRs M in a language L what are sufficient conditions for existence of an expansion of the language such that in the expanded M is still qRs and:

(i) there is a non-constant meromorphic function f : M → P(K)?

(ii) there are an algebraic curve C ⊆ P^{n}(K) and a biholomorphic (in a resonable sense) mapping f : M → C?

Remark In some cases (like in the example in section 5) a candidate for the function f (and f^{∗} for M^{∗}) are almost obvious. Then the problems above are equivalent to the conjuction of the two questions:

(1) is M ≺ M^{∗} in the language L ∪ {f}?

(2) is (M^{∗}, π) still e.c. in the language L ∪ {f}?

Example 4(C) If K is an ordered field, then K^{Γ} can be ordered. Let x = ∑ a_{γ}t^{γ} and y = ∑ b_{γ}t^{γ}. Then x < y if a_{g} < b_{g} where g is least such that a_{g} ≠ b_{g}. In the special case when K = R and Γ divisible the field R^{Γ} is real closed. This was considered in [DMM] in ”analytic language” L_{an}. The language includes, up to interdefinability, the field operations, the unary predicate I(x) saying |x| ≤ 1 and for any n-ary function f analytic (i.e. given by a power series) in a neighborhood of I^{n} there is a symbol f̃ ∈ L_{an} interpreted as f(x) for x ∈ I and f̃(x) = 0 if x ∉ I. It is proved that R_{an} ≺ R_{an}^{Γ}, where f̃ in R_{an}^{Γ} is interpreted by corresponding power series for |x| ≤ 1. There is an obvious way of defining C_{an} and C_{an}^{Γ} and interpreting the structures in R_{an} and R_{an}^{Γ} correspondingly (remember that a complex analytic function f of n variables x_{i} + iy_{i} is expressible in the form f = u + iv, where u and v are real analytic functions of 2n variables x_{i}, y_{j}. Also define I(C) = I(R)^{2}) ). Hence the results for R_{an} can be applied to this field too. Namely,

C_{an} ≺ C_{an}^{Γ}.

The Riemann Existence Theorem states that in the classical sense both (i) and (ii) of the problem above have positive answers. Now, given a classical compact Riemann surface M, one can interprete it in C_{an} since in the language we have complex discs I(C) with the analytic structure. More precisely, the language L for the complex manifold is two-sorted with sorts M and C. The language for sort C is L_{an}, the language for M contains finite number of unary predicates U_{i}, distinguishing discs covering M, and names for biholomorphic isomorphisms of U_{i} and I(C). Correspondingly, the same interpretation applied to C_{an}^{Γ} yields a structure M^{Γ}. Take now a meromorphic function f given by the Riemann Theorem and expand the language for M by a symbol for f with the appropriate interpretation of the symbol in M and in M^{Γ}. Then the fact that C_{an} ≺ C_{an}^{Γ} and Proposition 4 imply that f is meromorphic in the qRs sense and both (i) and (ii) have positive solutions.

4 Elements of analytic theory

We prove here some analogs of classic complex theory. We believe that it is possible to get much more complete analogy rewriting chapters of [GR] towards the following

Conjecture Any qRs M with respect to the family of analytic subsets of M^{n} (n ∈ N) is a Zariski structure (see [Z] and [HZ] for the definition).

We consider in this section a qRs M with all the structure defined in the preceding section and the topology C_{π}.

Lemma 4.1 (i) meromorphic functions on U ⊆ M^{n} are continuous; (ii) analytic subsets of U ⊆ M^{n} are closed in U.

Proof (i) is immediate from 1(ii) of the definition of O_{b}; (ii) follows from 1(i) and 1(ii) of the same definition.□

Lemma 4.2 Any proper analytic subset S of open connected U ⊆ M is discrete.

Proof Suppose not. Then the subset S_{lim} of limit points of S is not empty. Let a ∈ S_{lim}. This means there is a′ ∈ S^{∗} ∩ V_{a}, a′ ≠ a. Let ϕ be the defining germ of S at a. By the Weierstrass assumptions either ϕ = 0 or ϕ = x^{m} · g where g is a germ which does not vanish in V_{a} and x a parameter in a. In the second case ϕ(z) = 0 iff z = a hence only the first one is possible which means V_{a} ⊆ S^{∗}. This proves S_{lim} ⊆ Int(S). The opposite inclusion is immediate here and S_{lim} is closed, hence S_{lim} is clopen in U. By connectedness S_{lim} = U = S, the contradiction. □

Lemma 4.3 If f^{∗}(V_{a}) = {f(a)} for a meromorphic f on connected U ⊆ M and some a ∈ U then f is constant on U.

Proof The set of solutions of f(x) = f(a) is a nondiscrete analytic subset of U, hence by Lemma 4.2 is U. □

Proposition 10 For any nonconstant meromorphic function on M

f(M) = P(K).

Proof f(M) is closed since f is continuous and M quasi-compact. f(M) is open by Lemmas 3.1 and 4.3. Since P(K) is connected we have the equality. □

Definition For a meromorphic function f on U ⊆ M we say f has zero (pole) of order k in a ∈ U if f^{∗}|V_{a} = x^{k} · ϕ ( respectively 1/f^{∗}|V_{a} = x^{k} · ϕ ) where x is a parameter and ϕ an invertible germ.

Proposition 11 Two meromorphic functions on M differ by a nonzero constant multiplier if they have the same zeroes and poles with the same orders.

Proof If f_{1}, f_{2} are the functions then f_{1}/f_{2} has no zeroes and poles, thus by Proposition 10 must be a nonzero consant.□

Corollary 2 Any meromorphic function on P(K) is a ratio of two polynomials.

Lemma 4.4 Assume S ⊆ U is analytic and U ⊆ M^{n} open and connected. If the set of defining germs of S in some a ∈ U consists of constant zero germs then S = U.

Proof Let S_{0} = {a ∈ S : V_{a} ⊆ S^{∗}}. The set is non-empty by assumptions. Let b ∈ M, U_{b} = {u ∈ M^{n} : ⟨b, u⟩ ∈ U}, S(b) = {s ∈ M^{n−1} : ⟨b, s⟩ ∈ S}. Evidently ⟨b, c⟩ ∈ S_{0} implies S(b) = U_{b} by induction starting with Lemma 4.2. On the other hand the set

T = {b ∈ M : S(b) = U_{b}}

is an analytic subset of pr(U) the projection of U to the first coordinate since for b ∈ pr(U) the set of defining germs for T in b may consist of all germs of the form g(v, c) for all c ∈ U_{b} and g(v, w) defining S in ⟨b, c⟩. By our assumptions and Lemma 4.2 T = M. It follows S_{0} = M^{n} = S. □

Definition Let S ⊆ U ⊆ M^{n} be an analytic subset of an open U, a ∈ U. We say dim_{a}S ≤ m if a ∉ S or any set of defining germs in a consists of not less than n − m germs.

Definition An analytic subset S ⊆ U ⊆ M^{n} is said to be a hypersurface if it is non-empty and the set of defining germs consists of a one germ for any a ∈ U.

Lemma 4.5 Let ϕ ∈ O_{a}^{n} and x_{1} a parameter in O_{a}^{n} such that for any v ∈ V_{a} x_{1}(v) = 0 implies ϕ(v) = 0. Then x_{1} divides ϕ in O_{a}^{n}.

Proof Immediately from the Division Theorem for the rings of germs.

Lemma 4.6 Let U ⊆ M^{n}, S ⊆ U be a hypersurface, P the projection of S on the first coordinate, V the projection of U on the first coordinate. Then for any a ∈ P either

(a) P^{∗} ∩ V_{a} = V_{a} or

(b) P^{∗} ∩ V_{a} = {a} in which case for any b ∈ M^{n−1} if ⟨a, b⟩ ∈ S then the defining germ ϕ for S at ⟨a, b⟩ is of the form ϕ = x_{1}^{m} · h where x_{1} the parameter of O_{a}, m > 0 and h ∈ O_{⟨a,b⟩} does not vanish on V_{⟨a,b⟩}.

Proof Let ⟨a, b⟩ ∈ S, b = ⟨b_{2}, …, b_{n−1}, b_{n}⟩, ϕ a germ which defines S at ⟨a, b⟩. It follows from the Weierstrass assumptions that one can find m ≥ 0 such that ϕ = x_{1}^{m} · h where h is a germ not divisible by x_{1}. Suppose m > 0 and h⟨a, b⟩ ≠ 0. Then h is invertible and this implies case (b). On the other hand if m = 0 then h⟨a, b⟩ = 0. So if (a) does not hold we come to consider h ∈ M_{⟨a,b⟩}. Then by 4.5 h(a, v) is not identically 0 on V_{b} and thus, by the regularity assumption of (2iii), for this germ at b there are parameters x_{2}, … x_{n} of O_{b} such that h(a, v) is regular for x_{n} with respect to x_{2}, … x_{n}. This means just that for some b′ ∈ V_{b} x_{2}(b′) = … = x_{n−1}(b′) = 0 while h(a, b′) ≠ 0. Then the point ⟨a, b′⟩ witnesses that germ h of O_{⟨a,b⟩} is regular for x_{n} with respect to parameters x_{1}, x_{2}, … x_{n}. This implies by preparation

h = (x_{n}^{l} + f_{1}x_{n}^{l−1} + ⋯ + f_{l}) · g

where x_{n}^{l} + f_{1}x_{n}^{l−1} + ⋯ + f_{l} is the Weierstrass polynomial, f_{i} ∈ M_{⟨a,b⟩} ∩ O_{⟨a,b⟩}(x_{1}, … x_{n−1}), g ∈ O_{⟨a,b⟩} invertible. Now fix any a′ ∈ V_{a} and values t_{2}, …, t_{n−1} ∈ V_{0}. Let t_{1} = x_{1}(a′) and b′ be a point in V_{b} such that x_{2}(b′) = t_{2}, …, x_{n−1}(b′) = t_{n−1} which exists by assumption (3) of the definition of O_{b}. The choice of the values t_{1}, t_{2}, …, t_{n−1} of parameters x_{1}, … x_{n−1} fixes f_{1}(t_{1}, …, t_{n−1}), …, f_{l}(t_{1}, …, t_{n−1}). By algebraic closedness there is a value t_{n} ∈ K^{∗} for x_{n} which is a solution to

x_{n}^{l} + f_{1}(t_{1}, …, t_{n−1}) · x_{n}^{l−1} + ⋯ + f_{l}(t_{1} …, t_{n−1}) = 0.

Since π(t_{1}, …, t_{n−1}) = (0, … 0) one sees that t_{n} ∈ V_{0}. By assumption (3) for the ring of germs there is b′ ∈ V_{b} such that (x_{2}(b′), …, x_{n}(b′)) = (t_{1}, …, t_{n}), and hence h(a′, b′) = 0, thus ⟨a′, b′⟩ ∈ S^{∗}, that is a′ ∈ P^{∗} and (a) is satisfied. □

Proposition 12 Let S ⊆ M^{n} be a hypersurface, P its projection on the first coordinate. Then either P is finite or P = M.

Proof Since M is quasi-compact P is closed. Denote A the set of points a of M satisfying (b) of Lemma 4.6. For any a ∈ A the set S(a) = {s ∈ M^{n−1} : ⟨a, s⟩ ∈ S} is analytic and the defining germ of S(a) at b ∈ S(a) is by (b) of the form x_{1}(a, v), identically zero. Thus by Lemma 4.4 S(a) = M^{n−1}. It follows A is definable and analytic. Moreover, the set is discrete in P since A^{∗} ∩ V_{a} ⊆ P^{∗} ∩ V_{a} = {a} for any a ∈ A. Hence P \ A is closed. At the same time P \ A is open by Lemma 4.6. Thus either P = M or P \ A = ∅ and P = A. Since the last set is discrete and closed in M, P is finite in this case.□

5 One more example. A torus

A1. Take an algebraically closed field K and P(K)^{∗} ≻ P(K) saturated over P(K), π : P(K)^{∗} → P(K) a specialization with respect to Zariski closed relations. This determines the π-topology on P(K). We assume (π, P(K)^{∗}) is e.c. Denote K̇ = K \ {0}. This is an open subset of P(K) and carries also the group structure. A2. Let Ω be a multiplicative cyclic subgroup of K̇ generated by a transcendental element g.

Define M = K̇/Ω and consider closed those subsets S of M^{n} the inverse image of which are closed in K̇^{n}.

We show here that under some choice of π Ω becomes closed and discrete in K̇ and M inherits the structure of qRs from P(K). This is a direct analogy of the construction of 1-dimensional complex tori. It is probable that appropriate constructions of curves of higher genera using Schottky groups can be treated analogously. See [GvdP], [BGR] for ultrametric constructions of this type.

We have to start with some model theory of K in language

L_{Ω} = (+, ·, Ω, ≤_{Ω})

where

x ≤_{Ω} y iff (x = g^{k}&y = g^{l}&k ≤ l).

We also consider ≤_{Ω} a relation on the bigger set Ω̄ = Ω ∪ {0, ∞} ⊆ P(K) by putting 0 ≤_{Ω} x ≤_{Ω} ∞ for any x ∈ Ω. When we work in P(K) the graphs of · and + are extended to P(K) as ternary relations in canonical way.

Lemma 5.1 Suppose t_{1}, …, t_{k} ∈ Ω^{∗}, t = ⟨t_{1}, …, t_{k}⟩, v = ⟨v_{1}, …, v_{k}⟩, p(t) = 0 for p(v) a polynomial over Z[g]. Then there are monomials v^{i}, v^{l} of p (i, l are distinct multiindices) and an integer n such that t^{i} = g^{n}t^{l} holds. p = 0 is a consequence of the equations of the form v^{i_{j}} = g^{n_{j}}v^{l_{j}} which hold for

v = t.

Moreover, p = 0 modulo the condition Ω(v_{1})& … &Ω(v_{k}) is equivalent to a positive Boolean combination of equations of the form

v^{i_{j}} = g^{n}v^{l_{j}}.

Proof Let {v^{i} : i ∈ I(p)} be the monomials in v_{1}, …, v_{k} written in multiindices ordered lexicographically,

p(v) = Σ_{i∈I(p)}a_{i}v^{i}, a_{i} ∈ Z[g].

For every equation of the form v^{i} = g^{n}v^{l} (i < l) which holds for v = t make a reduction in p exchanging the first monomial by the second one. Denote p′ the polynomial resulting from all such reductions. Then t^{i} ≠ g^{n}t^{l} for all n and different multiindices i, l ∈ I(p′). Suppose, towards a contradiction, p′ is not trivial. Then, by the fact that Ω ≺ Ω^{∗}, there are h_{1}, …, h_{k} ∈ Ω, such that p′(h_{1}, …, h_{k}) = 0 and h^{i} ≠ g^{n}h^{l} for different i, l for all n such that |n| is not greater than the maximal power of g occuring in the coefficients a′_{i} of p′. Now as h^{i} are elements of Ω one can write them as h^{i} = g^{l(i)}. Thus Σ_{i∈I(p′)}a′_{i}g^{l(i)} = 0 is a non-trivial equation satisfied by g. The contradiction which proves that p = 0 is a consequence of the group equations. It is also proved that the set of solutions of p = 0 in Ω^{∗} is the union of solutions in Ω^{∗} of all the systems of semigroup-word equations constructed above. By Compactness Theorem there is a finite union equivalent to p = 0. □

Lemma 5.2 Suppose p(v_{1}, …, v_{k}) is a polynomial over K. Then there is finite number of polynomials p_{i}(v_{1}, …, v_{k}) over Z[g] such that p(v_{1}, …, v_{k}) = 0 is equivalent to ⋀_{i} p_{i}(v_{1}, …, v_{k}) = 0 modulo the condition Ω(v_{1})& … &Ω(v_{k}) and the v-monomials of p_{i} are the v-monomials of p.

Proof Let p(v) = Σ_{j∈J(p)}a_{j}v^{j} a_{j} ∈ K. Let H ⊆ Ω^{k} be the set of all solutions of p(v_{1}, …, v_{k}) = 0 in Ω. The system

(∗) Σ_{j∈J(p)}x_{j}h^{j} = 0 : h ∈ H

of homogeneous linear equations with coefficients in the subfield F (g) of K generated by g has a non-trivial solution in K and hence in F (g). Then, since the space of solution of (∗) is defined by linear equations over F (g), it has a basis with coordinates in Z[g] which we denote C, the strings of C denote ⟨c_{j,i} : j ∈ J(p)⟩. Put p_{i}(v) = Σ{c_{j,i}v^{j} : j ∈ J(p)}. Then the solutions of ⋀_{i} p_{i}(v_{1}, …, v_{k}) = 0 in F (g) are exactly those h ∈ F (g) for which ⟨h^{j_{1}}, …, h^{j_{m}}⟩ ({j_{1}, …, j_{m}} = J(p)) is orthogonal to C, which is the same as to be orthogonal to ⟨a_{j_{1}}, …, a_{j_{m}}⟩ which is equivalent to be a solution of p(v_{1}, …, v_{k}) = 0. □

Corollary 3 For any p(v_{1}, …, v_{k}) polynomial over K the equation p(v_{1}, …, v_{k}) = 0 modulo the condition Ω(v_{1})& … &Ω(v_{k}) is equivalent to a positive Boolean combination of equations of the form

v^{i_{j}} = g^{n_{j}}v^{l_{j}}

for some monomials v^{i_{j}}, v^{l_{j},} of p and integers n_{j}.

As a byproduct one has

Theorem 1 For Ω a multiplicative subgroup of K̇, generated by a transcendental elements g, the theory of ⟨K, +, ·, Ω⟩ is superstable.

Proof Consider any elementary extensions K_{1} of ⟨K, +, ·, Ω(K)⟩. Let X be a subset of K_{1}. By Corollary 3 there is a subset X′ ⊆ Ω(K_{1}) of the same power modulo ω as X such that any quantifier-free relation with parameters X between elements of Ω(K_{1}) is equivalent to a quantifier-free relation in the group language with parameters X′. Now consider any group-automorphism α of Ω(K_{1}) fixing X′. It follows from the above noticed that α can be extended to a field-automorphism ᾱ of the algebraic closure of Ω(K_{1}) ∪ X which fixes X. It is evident that any two elements outside the algebraic closure of Ω(K_{1}) ∪ X are conjugated by a field-automorphism which fixes the algebraic closure of Ω(K_{1}) ∪ X. It follows the number of types in the field language expanded by Ω and constants X is the same as the number of types over X′ in the group structure Ω(K_{1}). The latter structure is superstable, hence ⟨K, +, ·, Ω⟩ is. □

Proposition 13 For any elementary extension K′ ≻ K in language L_{Ω}, any polynomial p(v_{1}, …, v_{k}) over K′ the equation p(v_{1}, …, v_{k}) = 0 modulo the condition Ω(v_{1})& … &Ω(v_{k}) is equivalent to a positive Boolean combination P_{p}(v) of equations of the form

v^{i_{j}} = t_{j}v^{l_{j}}.

Proof By Corollary 3 this is true for K′ = K. Since the v-monomials in P_{p}(v) are the same as in p(v), given a degree d there is a finite number of positive Boolean combinations of equations of the form v^{i_{j}} = w_{j}v^{l_{j}} such that any polynomial over K of degree d is equivalent to one of the combinations after appropriately substituting w_{j} = t_{j} ∈ Ω(K) for all w_{j}. This can be written in the first order language, hence it is true in K′. □

Lemma 5.3 Suppose K′′ ≻ K^{∗} and both are elementary equivalent to K in language L_{Ω}, K′′ ⊇ K′ ⊇ K^{∗} as substructures in L_{Ω}. Then for any polynomial p over K^{∗} the equation p(v_{1}, …, v_{k}) = 0 modulo the condition Ω(v_{1})& … &Ω(v_{k}) is equivalent in K′ to a positive Boolean combination of equations of the form

v^{i_{j}} = t_{j}v^{l_{j}}

for some t_{j} ∈ Ω(K^{∗}).

Proof For the case K′′ = K^{∗} it is proved in Proposition 13. Consider a fixed polynomial p over K^{∗} and the appropriate t_{i} ∈ Ω(K^{∗}) such that the equivalence holds in the sense of K^{∗}. By definition this equivalence is true also for K′′ ≻ K^{∗} and thus for K′. □

Proposition 14 Let K^{∗} ≻ K in language L_{Ω}. Suppose there is given a cut X^{−}, X^{+} in the ordering of Ω^{∗} such that X^{−} has no maximal element and X^{+} has no minimal element. Then there is K′ ≻ K^{∗} in the language, such that Ω(K′) = Ω(K^{∗}) · ω^{Q}, where ω is an element transcendental over K^{∗}, X^{−} <_{Ω} ω <_{Ω} X^{+} and ω^{Q} is a minimal multiplicative subgroup containing ω and solutions of any equation of the form x^{n} = ω, n ∈ N.

Proof Take K′ = acl(K^{∗}(ω)) (acl in the field sense) for ω in Ω(K′′) for some large K′′ ≻ K^{∗} satisfying the type

X^{−} <_{Ω} ω <_{Ω} X^{+}& ⋀_{n∈N} ∃xΩ(x)&x^{n} = ω.

Then Ω(K′) ⊇ Ω(K^{∗}) · ω^{Q} by definition. On the other hand for any a ∈ acl(K^{∗}(ω)) there is a polynomial p(v_{1}, v_{2}) over K^{∗} such that p(ω, a) = 0. If Ω(a) holds then by Lemma 5.3 the equality is a consequence of a group-word equation with parameters in K^{∗} : ω^{k} · a^{l} = t for k, l ∈ Z, l ≠ 0, t ∈ Ω(K^{∗}). Hence Ω(K′) ⊆ Ω(K^{∗}) · ω^{Q}. By the elementary theory of ordered abelian groups

Ω(K′) ≻ Ω(K^{∗})

in language (·, ≤_{Ω}). Now consider a large saturated elementary extensions K_{1} of K^{∗} and K_{2} of K′ such that card(Ω(K_{2})) = card(Ω(K_{1})) and the transcendence degree of K_{2} over Ω(K_{2}) is equal to that of K_{1} over Ω(K_{1}). By the elementary ordered abelian groups equivalence there is an ordered group isomorphism β : Ω(K_{1}) → Ω(K_{2}) which fixes Ω(K^{∗}). Notice that Lemma 5.3 is applicable to the pair K′ ⊇ K^{∗} and thus to K_{2} ⊇ K^{∗}, and also it is applicable to K_{1} ⊇ K^{∗}. It follows that β can be extended to K^{∗} as identity. And finally β can be extended as a field isomorphism K_{1} → K_{2} which provides an isomorphism in language L_{Ω}. Since K^{∗} and K′ have a common elementary extension, K′ ≻ K^{∗}. □

Proposition 15 Suppose Ω is as in A2. Then one can choose π as in A1 such that A3. Ω is closed and discrete in K̇.

Proof Consider the compact Hausdorff topology on Ω̄ determined by the inequality ≤_{Ω} and the induced product topologies on Ω̄^{n}. By compactness of Ω̄ there is a specialization

π : Ω̄^{∗} → Ω̄

respecting ≤_{Ω} and ·. This specialization considered as a partial map P(K)^{∗} → P(K) respects the Zariski (field) structure on P(K), by Corollary 3. Extend now π to a total Zariski specialization P(K)^{∗} → P(K) by Proposition 1 and we are done, since π-topology coincides with the initial Hausdorff topology on Ω̄. □

Proposition 16 Consider the assumptions A1 - A3 above. Then M is quasi-compact.

Proof Define K^{#} = K^{∗} \ π^{−1}({0, ∞}). First we prove the following

Lemma 5.4 The mapping

i : K^{#}/Ω → (K̇/Ω)^{∗}

given as

i(α · Ω) = α · Ω^{∗}

is a bijection.

Proof The mapping is injective since Ω^{∗} ∩ K^{#} = Ω. It is surjective or, in other words, for any α ∈ K̇^{∗} there is an a ∈ K̇ and ω ∈ Ω^{∗} such that π(α · ω) = a. To see this we consider any α ∈ K̇^{∗} and look for a ∈ K̇ and ω ∈ Ω^{∗}.

If π(α) ∈ K̇ then ω = 1. So we assume π(α) = ∞. The case π(α) = 0 is symmetrical.

We look for ω in an extension of ⟨π, (P(K^{∗}), L_{Ω})⟩. We construct such an extension, ⟨π′, (P(K′), L_{Ω})⟩, assuming

π(α · γ) ∈ {0, ∞} for any γ ∈ Ω^{∗}.

This would contradict the fact that π is e.c. and prove Lemma. First define a cut in Ω^{∗} putting

X^{−} = {γ ∈ Ω^{∗} : π(γ · α) = 0}, X^{+} = {γ ∈ Ω^{∗} : π(γ · α) = ∞}.

These satisfy the assumptions of Proposition 14. Let K′ and ω be as Proposition 14 provides. Evidently placing of ω in the cut determines the ordering of Ω(K′) = Ω^{∗} · ω^{Q} uniquely. Since α·ω is transcendental over K^{∗}, the extension π′ to α·ω of the mapping π defined as

π′(α · ω) = 1

is a specialization with respect to the Zariski relations. By Proposition 1 this can be extended to a total Zariski-topology specialization

π′ : P(K′) → P(K).

Finally we show that π′ preserves ≤_{Ω} and Ω̄, then π′ will be the desired specialization. We call an element a ∈ Ω(K′) regular if it satisfies

a ≤_{Ω} g^{k} for all k ∈ Z iff π′(a) = 0 and a ≥_{Ω} g^{k} for all k ∈ Z iff π′(a) = ∞.

Notice that by definitions any element of Ω(K^{∗}) is regular. The end of the proof of Lemma is the proof of the regularity of all elements of Ω(K′). Claim 1. Any element of the form a = γ · ω with γ ∈ Ω^{∗} is regular. Proof By definitions a ≤_{Ω} g^{k} iff ω <_{Ω} g^{k}γ^{−1} iff ω <_{Ω} γ^{−1}. Also ω <_{Ω} γ^{−1} iff π(αγ^{−1}) = ∞. The latter can be rewritten as ∞ = π(αγ^{−1}) = π′(αω · ω^{−1}γ^{−1}) = π′(a^{−1}). Hence we come by equivalence to π′(a) = 0. If one starts with a > g^{k} one gets analogously the equivalence to π′(a) = ∞.

Claim 2. For any m, l ∈ Z, m ≠ 0, a is regular iff a^{m}g^{l} is. Proof If m > 0 then π′(a^{m}g^{l}) = 0 iff π′(a) = 0 and

a^{m}g^{l} ≤_{Ω} g^{k} for all k ∈ Z iff a ≤_{Ω} g^{k} for all k ∈ Z.

If m < 0 then π′(a^{m}g^{l}) = 0 iff π′(a) = ∞ and

a^{m}g^{l} ≤_{Ω} g^{k} for all k ∈ Z iff a ≥_{Ω} g^{k} for all k ∈ Z.

Claim 3. All elements of Ω(K′) are regular. Proof By Proposition 14 an arbitrary element a of Ω(K′) is of the form a = γ · ω_{1} where γ ∈ Ω(K′) and ω_{1}^{n} = ω^{m} for some n, m ∈ Z, n ≠ 0. By Claim 2 we can assume n = 1. We may also assume m ≠ 0 for otherwise a is regular. By elementary properties of Ω there are some δ ∈ Ω(K^{∗}) and l ∈ Z such that γ = g^{l}δ^{m}. Hence a = (δω)^{m}g^{l}. By Claim 1 the element in brackets is regular and so is a by Claim 2. This proves Lemma. □

Continuation of the proof of Proposition. We must find a specialization of (K̇/Ω)^{∗} onto K̇/Ω. By Lemma it is enough to find a specialization

σ : K^{#}/Ω → K̇/Ω.

It is evident that σ(α · Ω) = π(a) · Ω satisfies the required.□

Proposition 17 Suppose now P(K) is qRs with π satisfying Proposition 15. Then M = K̇/Ω is qRs.

Proof The canonical mapping K̇^{∗} → M^{∗} is bijective on infinitesimal neighborhoods, since Ω is discrete. Thus the mapping transfers in a natural way the rings of holomorphic germs on P(K) to M. This makes M qRs.□

Problem For arbitrary algebraically closed field is there a structure of qRs on P(K) which allows a non-constant meromorphic function

f : K̇/Ω → P(K)?

The theory of rigid spaces [BGR], [GvdP] gives a (unique in a sense) candidate, elliptic function, for f in the situation of a torus over algebraically closed field with a non-Archimedean valuation. We do not know its elementary properties. For example, consider F a graph of f as a function from K̇ and extend it to P(K) defining

⟨x, y⟩ ∈ F ≡ (f(x) = y ∨ x = 0 ∨ x = ∞).

It follows from qRs assumptions that the theory of P(K) in the Zariski language expanded by F must be atomic compact. Other questions on f where pointed at the end of section 3. REFERENCES

[B] V.G.Berkovich, Spectral Theory And Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, No 33, AMS, Providence, Rhode Island, 1990

[BGR] S.Bosch, U.Guentzer, and R.Remmert,Non-Archimedean Analysis, Grundlehren der Mathematischen Wissenschaften, Bd.261, Springer, Berlin-Heidelberg-New Yourk, 1984

[Bo] N.Bourbaki, Algébre Commutative, Hermann, Paris, 1961

[DMM] L.van den Dries, A.Macintyre, and D.Marker, The Elementary Theory of Restricted Analytic Fields with Exponentiation, to appear

[F] L.Fuchs, Partially ordered algebraic structures, Pergamon Press, 1963

[GR] R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Princeton Hall Inc., N.J.,1965

[GvdP] L. Gerritzen, M. van den Put, Schottky Groups and Mumford Curves, Lecture Notes in Math., v.817, Springer, Berlin-Heidelberg-New York, 1980

[HZ] E.Hrushovski and B.Zilber Zariski Geometries, Bull. AMS,28 (1993), 315-323

[R] P.Ribenboim, Théorie des valuations, Les Presses

[W] R.Walker, Algebraic Curves, Princeton University Press, Princeton, New Jersey, 1950

[Z] B.Zilber, Lecture notes on Zariski-type structures, Manuscript, 1992