Model theory of special subvarieties and Schanuel-type conjectures Boris Zilber 5 1 University of Oxford 0 2 February 26, 2015 b e F Abstract 5 We use the language and tools available in model theory to redefine 2 andclarify therather involvednotion of a special subvarietyknown from thetheory of Shimuravarieties (mixed and pure). ] G A 1 Introduction . h t 1.1 The first part of the paper (section 2) is essentially a survey of develop- a m mentsaroundtheprogramoutlinedinthetalktotheEuropeanLogicColloquium- 2000 and publication [29]. It then continues with new research which aims, on [ the onehandside, toextendthemodel-theoreticpicture of[29]andofsection2 2 to the very broad mathematical context of what we call here special coverings v of algebraic varieties, and on the other hand, to use the language and the tools 1 available in model theory to redefine and clarify the rather involvednotion of a 0 3 special subvarietyknownfromthe theoryofShimuravarieties(mixedandpure) 3 and some extensions of this theory. 0 Ourdefinitionofspecialcoveringsofalgebraicvarietiesincludessemi-abelian . 1 varieties, Shimura varieties (definitely the pure ones, and we also hope but do 0 not know if the mixed ones in general satisfy all the requirement) and much 5 more, for example, the Lie algebra covering a simple complex of Lie group 1 SL(2,C). : v Recall from the discussion in [29] that our specific interest in these matters i arosefromthe connectionto Hrushovski’sconstructionofnewstablestructures X (seee.g. [19])andtheirrelationshipwithgeneralisedSchanuelconjectures. This r a subject is also closely related to the Trichotomy Principle and Zariski geome- tries. Inthecurrentpaperweestablishthatthegeometryofanarbitraryspecial covering of an algebraic variety is controlled by a Zariski geometry the closed subsets of which we call (weakly) special. The combinatorial type of simple (i.e. stronglyminimal)weaklyspecialsubsetsareclassifiablebytheTrichotomy Principle. Using this geometry and related dimension notions we can define a correspondingverygeneralanalogueof“Hrushovski’spredimension”andformu- late corresponding“generalisedSchanuel’s conjecture”as wellas a verygeneral formsofAndr´e-Oort,theCITandPink’sconjectures(Zilber-Pinkconjectures). Notethatinthisgeneralityonecanseeaconsiderableoverlapofthegeneralised Schanuelconjectures with the Andr´econjecture onperiods [1] (generalisingthe 1 Grothendieck periodconjecture)which promptfurther questionson the model- theoretic nature of fundamental mathematics. 1.2 Acknowledgements. The idea of approaching the most general setting of Schanuel-type conjectures crystallised during my 4-weeks visit at IHES at Bures-Sur-Yvette in February 2012, and considerable amount of work on this projectwascarriedout during my 2-monthsstay atMPIM, BonninApril-May 2012. My sincere thanks go to these two great mathematical research centres and people with whom I met and discussed the subject there. Equally crucial impact on this work comes from many conversations that I had with Jonathan PilawhoseworkondiophantineproblemsaroundShimuravarietiesusingmodel- theoretic tools introduced me to the subject. Moreover,the success of his work and of the Pila-Wilkie method brought together people working in different areasand made it possible for me to consult E.Ullmo, A.Yafaev, B.Klinger and others on the topic. On model-theoretic side I used Kobi Peterzil’s advice on o-minimality issues. My thanks go to all these people. 2 Analytic and pseudo-analytic structures RecallthatastronglyminimalstructureM(oritstheory)canbegivenacoarse classification by the type of the combinatorial geometry that is induced by the pregeometry (M,acl) on the set [M \acl(∅)]/ , where x∼y iff acl(x)=acl(y). ∼ The Trichotomy conjecture by the author stated that for any strongly min- imal structure M the geometry of M is either trivial or linear (the two united under the name locally modular), or the geometry of M is the same as of an algebraically closed field, and in this case the structure M is bi-interpretable with the structure of the field. E.Hrushovskirefutedthisconjectureinthegeneralsetting[19]. Nevertheless the conjecture was confirmed, by Hrushovski and the author in [20], for an im- portant subclass of structures, Zariski geometries, except for the clause stating the bi-interpretability, where the situation turned out to be more delicate. 2.1 Recall that the main suggestion of [29] was to treat an (amended version of) Hrushovski’s counter-examples as pseudo-analytic structures, analogues of classicalanalyticstructures. Hrushovski’spredimension,andthe corresponding inequality δ(X) ≥ 0, a key ingredient in the construction, can be seen then to directly correspond to certain type of conjectures of transcendental number theory, which we called Generalised Schanuel conjectures. The ultimate goalin classifying the above mentioned pseudo-analyticstruc- tureshasbeentogivea(non-first-order)axiomatisationandproveacategoricity theorem for the axiomatisable class. 2.2 Thealgebraicallyclosedfieldswithpseudo-exponentiation,F =(F;,+,·,exp), exp analogues of the classical structure C = (C;,+,·,exp), was the first class exp studied in detail. The axioms for F are as follows. exp ACF : F is an algebraically closed fields of characteristic 0; 0 2 EXP: exp:G (F)→G (F) a m is a surjective homomorphismfromthe additive groupG (F) to the multiplica- a tive group G (F) of the field F, and m kerexp=ωZ, for some ω ∈F; SCH: for any finite X, δ(X):=tr.deg (X ∪exp(X))−ldim (X)≥0. Q Q Here tr.deg(X) and ldim (X) are the transcendence degree and the dimension Q of the Q-linear space spanned by X, dimensions of the classical pregeometries associated with the field F, and δ(X) takes the role of Hrushovski’s predimen- sion,whichgivesrisetoanewdimensionnotionandnewpregeometryfollowing Hrushovski’s recipe. The inequality can be recognised as the Schanuel conjec- ture, if one also assumes F =C . exp exp Since F results from a Hrushovski-Fr¨aisse amalgamation, the structure exp is Existentially Closed with respect to the embeddings respecting the predi- mension. This takes the form of the following property. EC: For any rotund and free system of polynomial equations P(x ,...,x ,y ,...,y )=0 1 n 1 n there exists a (generic) solution satisfying y =expx i=1,...,n. i i The term rotund has been coined by J.Kirby, [22], rotund and free is the same as normal and free in [30]. We refer the reader to [22] or [30] for these technical definitions. Finally we have the following Countable Closure property. CC:Formaximalrotundsystemsofequationsthe setofsolutionsisatmost countable. The main result of [30] is the following. 2.3 Theorem. Given an uncountable cardinal λ, there is a unique model of axioms ACF + EXP + SCH + EC + CC of cardinality λ. 0 2.4 Recall that in [29] we stated: Conjecture. C is isomorphic to the unique model F of axioms exp exp ACF + EXP + SCH + EC + CC, of cardinality continuum. 0 P. D’Aquino, A.Macintyre and G.Terzo embarked on a programme of com- paring F with C , in particular checking the validity of (EC) which they exp exp renamed the Nullstellensatz for exponential equation. The paper [14] proves that F contains a solution to any equation f(z) = 0, where f(z) is a one- exp variableterminexp,+and×withparameters,providedf(z)isnotoftheform exp(g(z)) for a term g(z). This was proved to hold in C by W. Henson and exp L. Rubin using Nevanlinna Theory, [18]. A. Schkop in [28] arrives at the same statement for F , deriving it directly from the axioms. exp In [13] P. D’Aquino, A.Macintyre and G.Terzo find a context in which the Identity Theorem of complex analysis can be checked and confirmed in F . exp 3 TheIdentityTheoremsaysinparticularthatifthezerosetofanentirefunction f(z) has an accumulationpoint then f ≡0. Since F comes with no topology exp there is no direct counterpart to the notion of accumulation in it. Instead [13] considers specific subsets of F , such that Q or the torsion points where an exp accumulation point can be defined in the same way as in C. In such cases, for a certain type of functions f D’Aquino, Macintyre and Terzo prove that the conclusion of the Identity Theorem holds in F . It is interesting that the exp results are obtained by invoking deep theorems of Diophantine Geometry. In the same spirit the present author tried, without much success, to solve the following: Problem. Describesyntacticallytheclassofquantifier-freeformulaeσ(z ,...,z ) 1 n in the language {exp,+,×} with parameters in C for which the subset S ={a¯∈Cn :σ(a¯)} is analytic in C. Note that the class is bigger than just the class of positive quantifier-free formulae. E,g,the following formula defines an entire function on C (and so its graph is analytic): exp(x)−1, if x6=0, y = x e, otherwise. (cid:26) 2.5 This is a consequence of Theorems A and B. Theorem A. The L (Q)-sentence ω1,ω ACF + EXP + SCH + EC + CC 0 isaxiomatisingaquasiminimalexcellentabstractelementaryclass(AEC). Theorem B. A quasiminimal excellent AEC has a unique model in any uncountable cardinality. 2.6 The original definition of quasiminimal excellence and the proof of Theo- rem B is in [31] based on earlier definitions and techniques of S.Shelah. Definition. Let M be a structure for a countable language, equipped with a pregeometry cl. We say that the pregeometry of M is quasiminimal if the following hold: 1. The pregeometry is determined by the language. That is, if tp(a,¯b) = tp(a′,¯b′) then a∈cl(¯b) if and only if a′ ∈cl(¯b′). 2. M is infinite-dimensional with respect to cl. 3. (Countable closureproperty)If X ⊆M is finite, then cl(X)is countable. 4. (Uniqueness of the generic type) Suppose that H,H′ ⊆M are countable closed subsets, enumerated such that tp(H) = tp(H′). If a ∈ M \H and a′ ∈ M \H′ then tp(a,H)=tp(a′,H′). 5. (ℵ -homogeneity over closed sets (submodels) and the empty set) Let 0 H,H′ ⊆M becountableclosedsubsetsorempty,enumeratedsuchthattp(H)= tp(H′), ), let ¯b,¯b′ be finite tuples from M such that tp(¯b,H) = tp(¯b′,H′), and let a∈cl(H,¯b). Then there is a′ ∈M such that tp(a,¯b,H)=tp(a′,¯b′,H′). Excellenceofaquasiminimalpregeometryisanextraconditionontheamal- gams of independent systems of submodels which we do not reproduce here in the general form but going to illustrate it in examples below. 4 TheproofofTheoremAreliesonessentialalgebraicanddiophantine-geometric facts and techniques and goes through an intermediate stage which is the fol- lowing. Theorem A .The natural L -axiomatisation of the two-sorted structure 0 ω1,ω (G (C),exp,C ) defines a quasiminimal excellent AEC. a field Here C is the field of complex numbers, and exp is the classical homo- field morphism exp:G (C)→G (C) (1) a m of the additive group onto the multiplicative group of complex numbers. The natural axiomatisation consists of the first-order part which consists of the theory of C , the theory of G (C) and the statement that exp is a field a surjective homomorphism. The only proper L -sentence which is added to ω1,ω this states that the kernel of exp is a cyclic group. These proofs were not without errors. The original paper [32] of A es- 0 tablished quasiminimality of (G (C),exp,C ) but has an error in the part a field provingexcellence. This wascorrectedin[2], where alsoageneralisationofthis theorem to a positive characteristic analogue was given. But after [2] a related errorintheproofofthemainTheoremAstillrequiredanextraargumentwhich did arrive but from an unexpected direction. M.Bays and J.Kirby in [3] (us- ing [21]), followed by M.Bays, B.Hart, T.Hyttinen, M.Kesaala and J.Kirby [4], andfurtheronfollowedbyL.Haykazyan[17]foundanessentialstrengtheningof Theorem B, which made certain algebraic steps in the proofs of Theorems A 0 and A redundant. The final result, see [4], is the following. 2.7 Theorem B∗. A quasi-minimal pregeometry can be axiomatised by an L (Q)-sentence which determines an uncountably categorical class. In par- ω1,ω ticular, the class is excellent. With the proof of this theorem the proof of the main theorem 2.3 has been completed. Theorem B∗ by itself is a significant contribution to the model theory of abstract elementary classes which, remarkably, has been found while working on applications. The significance of this model-theoretic theorem will be further emphasised in the discussion of its implications for diophantine geometry below. 2.8 Analogues of TheoremA are now established for elliptic curves,M.Bays, 0 and Abelian varieties by M.Bays, B.Hart and A.Pillay, based on earlier contri- butions by M.Gavrilovich,M.Bays and the author. Theorem A (M.Bays, [5]). Let E be an elliptic curve without complex Ell multiplicationoveranumberfieldk ⊂C.ThenthenaturalL -axiomatisation 0 ω1,ω ofthetwo-sortedstructure(G (C),exp,E(C))definesanuncountablycategorical a AEC. In particular, this AEC is excellent. Here exp : G (C) → E(C) is the homomorphism onto the group on the a elliptic curve given by the Weierstrass function and its derivative. The natural axiomatisation includes an axioms Weil which fixes the polynomial relation m between two torsion points of order m for a certain choice of m. If this (first- order) axiom is dropped, the categoricity fails but not gravely – the resulting 5 L -sentence still has only finitely many (fixed number of) models in each ω1,ω uncountable cardinality. The following is an extension of the previous result to abelian varieties in- corporated in [6] by M. Bays, B. Hart and A. Pillay Theorem A . Let A be an Abelian variety over a number field k ⊂ AbV 0 C and such that every endomorphism θ ∈ O (complex multiplication) is de- fined over k . Then the naturalL -axiomatisation of the two-sorted structure 0 ω1,ω (Cg ,exp,A(C)) along with the first-order type of the kernel of exp in the O·mod two-sorted language defines an uncountably categorical AEC. In particular, this AEC is excellent. Here Cg is the structure of the O-module on the covering space Cg, O·mod where g =dimA. We explain the main ingredients of the proof. 2.9 Definition. Let F be an algebraically closed field of countable tran- scendence degree and B a finite (possibly empty) subset of its transcendence basis. An independent system of algebraically closed fields is the collection L={L ⊆F:s⊆B} of algebraically closed subfields, L =acl(s). s s The boundary (or the crown) ∂L of L is the field generated by the L , s s6=B, ∂L=hL :s(Bi. s Nowletusalsoconsiderasemi-AbelianvarietyAoveranumberfieldk and 0 assume that F is of characteristic 0, k ⊂F. 0 For any subfield k ⊆ k ⊆ F, the set A(k) of k-rational points of A is well- 0 defined. And conversely, for D ⊆ A(F) we write k (D) the extension of k by 0 0 (canonical) coordinates of points of D. TheA-boundary∂ LofthesystemListhecomplexmultiplicationsubmod- A ule ∂ L=hA(L ):s(Bi+Tors(A), A s where Tors(A) is the torsion subgroup of A. The extension k =k (Tors(A)) will be of importance below. ∞ 0 We also needto use the Tate module T(A)ofA whichcanbe definedas the limit T(A)=limA m ← of torsion subgroup A of A of order m. m Given σ ∈Gal(F:k ) and a∈A(k ) define, for n∈N, ∞ ∞ hσ,ai =σb−b∈A , n n for an arbitrary b∈A such that nb=a. This does not depend on the choice of b and taking limits we have the map h ·, ·i : Gal(F:k )×A(k )→T(A), hσ,ai =limhσ,ai . ∞ ∞ ∞ ∞ n ← This also works then for k ⊇ k in place of k and we can consider, given ∞ ∞ a∈A(k), the submodule hGal(F:k),ai of the Tate module. ∞ ThemainingredientoftheproofofTheoremsA andA isthefollowing. 0 AbV 6 Theorem (“Thumbtack Lemma”). Let D be an A-boundary, let k be a finitely generated extension of k (D) and let a∈A(k) be such that O·γ∩D = 0 {0}. Then hGal(F:k),ai is of finite index in T(A). ∞ This splits naturally into 3 cases depending on the size n := |B|, namely cases n=0, n=1 and n>1. Thecasen=0essentiallyfollowsfromacombinationofafinitenesstheorem byFaltingsandtheBashmakov-RibetKummertheoryforAbelianvarietiesjust recently finalised by M.Larsen [25]. The similar result needed for the proof of TheoremA forG isjustDedekind’stheoryofidealsandtheclassicalKummer 0 m theory. Thecasen=1isthefieldoffunctionscaseoftheMordell-Weiltheoremdue to Lang and N´eron, see [24], Theorem 6:2. Thecasen>1isnewandrequiredaspecialtreatment. ForG itwasdone m in [2] using the theory of specialisations (places) of fields, but [6] finds a more directmodel-theoreticargumentwhichcoversalsothe caseofAbelianvarieties. Using the new model-theoretic result above one has now the following, for all reasonable forms of the Thumbtack Lemma (see, in particular, 2.14). Corollary to Theorem B∗. The case n > 1 in the Thumbtack Lemma follows from the cases n=0,1. 2.10 Recall that the statement of Theorem A is weaker than that of The- AbV orems A and A in including the complete description of the kernel of exp 0 Ell in the two-sorted language. The stronger version requires an extension of the Thumbtack Lemma which includes the case n = 0 & γ ∈ Tors. In other words one needs to characterise the action of Gal(k˜:k) on Tors. For G this is given m by the theory of cyclotomic extensions and for elliptic curves without complex multiplication by Serre’s theorem. For the general Abelian variety this is an open problem. Onthe other hand Bays,Hart andPillay in[6] provea broadgeneralisation of A-style theorems at the cost of fixing more parameters in the “natural ax- iomatisation”. Their axioms include the complete diagramof the prime model. In this setting Theorem A holds for an arbitrary commutative algebraic group over algebraically closed field of arbitrary characteristic. In fact, [6] building up on [7] by Bays, Gavrilovich and Hils shows how to generalise the statement to an arbitrary commutative finite Morley rank group and proves it in this formulation. Intermsofthe ThumbtackLemma thelatter requiresonlyn=1case. This is essentially Kummer theory over function fields in its most general, in fact model-theoretic, form. Bays, Gavrilovich and Hils in [7] use this technique for an application in algebraic geometry. 2.11 Note a version of Theorem A formulated in [6]. AbV Theorem. Modelsofthenaturalfirstorderaxiomatisationof(Cg ,exp,A) O·mod are determined up to isomorphism by the transcendence degree of the field and the isomorphism type of the kernel. This is a model-theoretic “decomposition” statement for the rather com- plex algebraic structure, similar to the Ax-Kochen-Ershov “decomposition” of henselian valued fields into residue field and value group. 7 2.12 Thestudyoftheabovepseudo-analyticstructuresshedsomelightonclas- sical transcendental functions, namely the complex exponentiation exp (Theo- rem A, 2.3), the Weierstrass function P(τ ,z) as a function of z (Theorem 0 A ) and more generally Abelian integrals and the corresponding exponentia- Ell tion (Theorem A ). Although a lot of questions remain still open, especially AbV forthelatter,anaturalcontinuationofthisprogramleadstoquestionsonmodel theory of P(τ,z) as a function of two variables z, and similar multy-variable maps related to Abelian varieties. But before anything could be said about P(τ,z) the classical function j(τ), the modular invariant of elliptic curves E , which can be defined in terms of τ P(τ,z). 2.13 Thetwo-sortedsettingforj(τ)analogoustosettingsin2.8isthestructure (H,j,C ), where H is the upper half-plane as a GL+(2,Q)-set, that is with field a b the action by individual elements from the group (rational matrices c d (cid:18) (cid:19) with positive determinants) a b aτ +b :τ 7→ . c d cτ +d (cid:18) (cid:19) In the language we have names for fixed points t ∈ H of transformations g g, which are exactly the quadratic points on the upper half-plane, and the list of statements gt = t is part of the axiomatisation. The images j(t ) ∈ C g g g are called special points. They are algebraic and their values are given by the axioms of the structure. The natural axiomatisation of this structure states that j : H → C is a surjection such that for every g ,...,g ∈ GL+(2,Q) there is an irreducible 1 n algebraic curve C ⊂Cn+1 over Q(S), where S is the set of special points, g1...gn such that hy ,y ,...,y i∈C ⇔∃τ ∈H y =j(τ),y =j(g τ),...y =j(g τ). 0 1 n g1...gn 0 1 1 n n This is given by a list of first-order sentences. Finally, the L -sentence ω1,ω j(τ)=j(τ′)⇔ τ′ =gτ g∈S_L(2,Z) states that the fibres of j are SL(2,Z)-orbits. Adam Harris proves in [16] an analogue of Theorems A above. 2.14 Theorem Aj The natural axiomatisation of (H,j,C ), the two-sorted field structure for the j-invariant, defines an uncountably categorical AEC. The structure of the proof is similar to the proofs discussed above. The key model-theoretic tool is Theorem B∗. The appropriate thumbtack lemma takes the following form. Theorem Let A be an abelian variety defined over k, a finitely generated extension of Q or a finitely generated extension of an algebraically closed field L, such that A is a product of r non-isogenous elliptic curves (with j-invariants which are transcendental over k in the second case). Then the image of the Galois representation on the Tate module T(A) is open in the Hodge group Hg(A)(Zˆ). 8 The Hodge group is a subgroup of the Mumford-Tate group, for definitions see e.g. [11]. This theorem, essentially a version of the adellic Mumford-Tate conjecture, is explained in [16] as a direct consequence of a version of the hard theorem of Serre mentioned above and a further work by Ribet. Consequently Harris deduces the categoricity theorem Aj. Whatisevenmorestriking,thatassumingthestatementoftheoremAj does hold, [16] deduces the statement of the adellic Mumford-Tate conjecture as a consequence. Thissortofequivalencewasobservedin[34]forcategoricitystate- ments for semi-Abelian varieties. The case considered by Harris, the simplest case of a Shimura variety, looks very different. And yet a similar tight connec- tion betweenthe modeltheory andthe diophantine geometry ofthe j-invariant is valid. 2.15 Most recent paper [12] by C.Daw and A.Harris extends Theorem Aj to much broader class of two sorted structures, replacing C by an arbitrary field modular curve and j by the corresponding modular function. Moreover,C.DawandA.Harrisextendtheformalismofthetwo-sortedstruc- turetoanyShimuravarietyandshowthattherespectivecategoricitystatement is equivalentto an(unknown)openimage conditiononcertainGaloisrepresen- tations. This equivalence between model-theoretic statements and those from arithmetic may be seen as a form of justification for the latter. 2.16 We would like to remark that in terms of the discussion in 2.10, 2.11 the axiomatisation 2.13 assumed in Theorem Aj is not the most natural one. Its language includes constants naming special points, as a result of which the actionofGal(Q˜ :Q)onspecialpoints isnotseeninthe automorphismgroupof the structure. Working in a more basic language one would require a stronger versionofthecorresponding“ThumbtackLemma”andthusadeeperstatement of Hodge theory. 2.17 A natural question (also asked by the anonymous referee) arises at this point: whatwouldtheanalogueofTheorem2.3(one-sortedcase)be inthecase of j in place of exp, or even in in the context of 2.8, for elliptic curves and Abelian varieties? Thecaseofanellipticcurveisexpectedtobequitesimilarto2.3. Oneneeds justtotryandprovethecategoricitystatementforthestructure(C;,+,·,P(τ,z)), where τ is the modular parameter of the elliptic curve in question and P(τ,z) is the Weierstrass function in variable z. The abelian variety case shouldn’t be very different; just replace P(τ,z) by the g functions p (z¯),...,p (z¯) of vari- 1 g ables z¯ on Cg, g the genus of the variety, which make up the exponentiation map from Cg onto the variety. The necessary form of the generalised Schanuel condition is known (but not the proof of it!), see the discussion in 2.23. Incaseofjthereisanessentialdifference: thenaturaldomainofdefinitionof j isH,theupperhalfplaneofthecomplexnumbers. Since(C;+,·,H)interprets the reals, (C;+,·,j) can not be excellent. Hence one needs to look for an extensionofjtothecomplexplaneminusacountablesetinawaythatpreserves its modularity andthe correspondinggeneralisedSchanuelcondition(see 2.23). Thiswouldbeverydesirablefornumbertheoreticreasons,seeYu.Manin’spaper [26] where a related problem is discussed. An easy argument shows that this is impossible. 9 Still, we may speculate, a possible solution to the problem would be to consider a “multivalued function” J instead of j. More precisely, define J(x,y) to be a binary relation on C such that for x∈H, y ∈C J(x,y) iff ∃g ∈GL+(Q) y =j(gx). 2 We want to extend J to C so that for all x∈C\Q, y ∈C and g ∈GL (Q), 2 J(x,y)→J(gx,y), and for all x∈C\Q, y ,y ∈C 0 1 J(x,y ) & J(x,y )→∃g ∈GL+(Q) hy ,y i∈C , 0 1 2 0 1 g where C is as in 2.13. g For such a J the same generalised Schanuel condition SCH as for j makes J sense. Now it is possible to formulate EC and have the analogue of 2.3 as the J conjecture for the one-sorted structure (C;+,·,J). The evidence supporting such a conjecture can be seen in the recently (for- mulatedand)provedAx- Schanuelstatementforthej invariantinthedifferen- tialfield setting,[27], Conjecture 7.12,the proofby J.TsimmermannandJ.Pila to appear elsewhere. The tightrelationshipbetweenthe statementof2.3, the theoremofAx (the Ax - Schanuel statement) and the model theory of differential fields has been given a deep analysis in the work of J.Kirby [22]. It should be also applicable in the case in question. 2.18 Finally, we would like to discuss the first-order versus non-first-order (AEC) alternative in choosing a formalism to develop the model theory of pseudo-analytic structures as above. InthisregardthereisasubstantialdifferencebetweenresultsoftypeA(The- orems A) and the main result about the “one-sorted” pseudo-exponentiation stated in 2.3. In the first situation, as shown in [6], one can use essentially the same techniques to classify models of the first order theory of the two-sorted structure in question. In the one-sortedcase the key property on which the whole model-theoretic study relies is Hrushovski’s inequality (recall 2.1) or in more concrete form the relevantSchanuel’scondition. The“decomposition”approachasin2.11canstill beattemptedbut,aswasnotedalreadyin[29],toformulateSchanuel’scondition in the first order way one requires certain degree of diophantine uniformity, whichwasformulatedin[35]astheConjectureonIntersectionsinTori,CIT.[29] discusses a broader formulation of this which includes semi-Abelian varieties. AnequivalenttoCIT conjecturewaslaterformulatedbyE.Bombieri,D.Masser and U.Zannier, and a very general form, which covers the whole class of mixed Shimura varieties, formulated by R.Pink. Without giving precise definition of special subvarieties (the secondpartof this paper is devotedto this, see section 6) we formulate what is currently referred to as the Zilber-Pink conjecture. 2.19 Conjecture Z-P. Let X be an algebraic variety for which the notion of specialsubvarieties is well-defined. Foranyalgebraic subvarietyV ⊆Xn thereis 10