A differential approach to the Ax-Schanuel, I
aa r X i v : . [ m a t h . N T ] F e b A DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I
DAVID BL ´AZQUEZ-SANZ, GUY CASALE, JAMES FREITAG, AND JOEL NAGLOO
Abstract.
In this paper, we prove several Ax-Schanuel type results for uniformizers ofgeometric structures. In particular, we give a proof of the full Ax-Schanuel Theorem withderivatives for uniformizers of any Fuchsian group of the first kind and any genus. Ourtechniques combine tools from differential geometry, differential algebra and the modeltheory of differentially closed fields. The proof is very similar in spirit to Ax’s proof of thetheorem in the case of the exponential function. Introduction
In this paper we use techniques from differential geometry, differential algebra and themodel theory of differentially closed fields to prove several results of Ax-Schanuel type. Ourmain result is stated in the context of a G -principal bundle π : P → Y with Y a complexalgebraic variety and G an algebraic group. Theorem A.
Let ∇ be a G -principal connection on P → Y with Galois group Gal ( ∇ ) = G .Let V be an algebraic subvariety of P and L an horizontal leaf. If dim V < dim( V ∩ L ) +dim G and dim( V ∩ L ) > then the projection of V ∩ L in Y is contained in a ∇ -specialsubvariety. Here, by a ∇ -special subvariety X ⊂ Y , we mean a subvariety such that Gal( ∇| X ) isa strict subgroup of G = Gal ( ∇ ). We use Theorem A to prove several functional tran-scendence results for uniformizers of a ( G, G/B )-structure on an algebraic variety Y . Theclassical and most studied examples of such structures come from Shimura varieties. Take G to be a connected semi-simple algebraic Q -group. Then for K , a maximal compact sub-group of G ( R ), we have that Ω = G ( R ) /K is a bounded symmetric domain. It is knownthat the compact dual ˇΩ of Ω is given as the quotient ˇΩ = G ( C ) /B for a Borel subgroup B of G . This quotient ˇΩ is a homogeneous projective variety and Ω is a semi-algebraicsubset (if we assume K ⊂ B ). Given an arithmetic lattice Γ ⊂ G ( Q ), the analytic quo-tient Y := Γ \ Ω = Γ \ G ( R ) /K has the structure of an algebraic variety and is called a(pure) Shimura variety. As detailed in Subsection 3.2, a ( G, G/B )-structure on Y can takento be the system of partial differential equations satisfied by a uniformization function q : Ω → Y := Γ \ Ω.We will work, in this paper, with more general (
G, G/B )-structures. For example, ourtheory allows for quotients of Ω (as above) by an arbitrary lattice Γ ⊂ G ( R ). Furthermore, it Date : February 9, 2021.2010
Mathematics Subject Classification. also applies to the differential equations satisfied by conformal mappings of circular polygons[16, Chapter 4]. In any case, we will show that attached to any (
G, G/B )-structure is a G -principal connection ∇ and so are able to apply Theorem A and obtain the followingresult. Corollary B.
Let υ be a uniformization of an irreducible ( G, G/B ) -structure on an alge-braic variety Y . Assume W ⊂ G/B × X is an irreducible algebraic subvariety intersectingthe graph of υ . Let U be an irreducible component of this intersection such that dim W < dim U + dim X, dim U > . Then the projection of U to X is contained in a ∇ -special subvariety of X . As we shall later see in the paper, many more results naturally follow from Theorem A.Moreover, if we assume in addition that the (
G, G/B )-structure is simple (see Definition 5.3),a natural assumption, then we are also able to obtain an Ax-Schanuel Theorem (includingwith derivatives) for products of Y . This gives a result which has a slightly weaker conclusionthan the general case of the main theorems of [27] and [10] but which applies to more generalsituations. Theorem C.
Let ( Y, Y ) be a simple ( G, G/B ) -structure on Y , ˆ t . . . ˆ t n be n formal param-eterizations of (formal) neighborhoods of points p , . . . p n in G/B and υ , . . . , υ n be solutionsof Y defined in a neighborhood of p , . . . p n respectively. If tr . deg . C C (cid:16) ˆ t i , ( ∂ α υ i )(ˆ t i ) : 1 ≤ i ≤ n, α ∈ N dim Y (cid:17) < dim Y + n dim G then there exist i < j such that tr . deg . C C ( υ i (ˆ t i ) , υ j (ˆ t j )) = tr . deg . C C ( υ i (ˆ t i )) = tr . deg . C C ( υ j (ˆ t j )) = dim G. Theorem C does not (in full generality) give any details about the kinds of special sub-varieties (or correspondences) that can occur - a problem we will tackle in general in sequelto this paper. Nevertheless, it will be crucial in giving a model theoretic analysis of therelevant partial differential equations. For example, as a consequence we are able to showthat the sets Y defined by the equations (along with some natural inequalities), in a differ-entially closed field, are strongly minimal and geometrically trivial. Building on this modeltheoretic analysis, we are able to give a complete analysis in the case of hyperbolic curves.Let Γ ⊂ PSL ( R ) be a Fuchsian group of the first kind and let j Γ be a uniformizing functionfor Γ. Notice here that G = PSL , Ω = H and ˇΩ = CP . Let ˆ t , . . . , ˆ t n be formal parameter-izations of neighborhoods of points p , . . . p n in H . We write δ i for the derivations inducedby differentiation with respect to ˆ t i . We prove the Ax-Schanuel Theorem with derivativesfor j Γ : Theorem D.
Assume that ˆ t , . . . , ˆ t n are geodesically independent, namely ˆ t i is nonconstantfor i = 1 , . . . , n and there are no relations of the form ˆ t i = γ ˆ t j for i = j , i, j ∈ { , . . . , n } and γ is an element of Comm (Γ) , the commensurator of Γ . Thentr.deg. C C (ˆ t , j Γ (ˆ t ) , j ′ Γ (ˆ t ) , j ′′ Γ (ˆ t ) , . . . , ˆ t n , j Γ (ˆ t n ) , j ′ Γ (ˆ t n ) , j ′′ Γ (ˆ t n )) ≥ n + rank ( δ i ˆ t j ) . Notice that the statement given above, in the spirit of Ax’s original paper, is slightlystronger than the one found for example in [31] whereby t , . . . , t n are assumed to be the DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 3 coordinate functions of some complex analytic subvariety of some open subset D ⊂ H n .Besides being a stronger result than the Ax-Lindemann-Weierstrass theorem from [14], italso generalizes the setup by dropping the assumption that the quotient is genus zero.1.1. Applications of Ax-Schanuel theorems.
Over the last decade, functional transcen-dence results, often in the form of the Ax-Lindemann-Weierstrass type results for certain an-alytic functions have played a key role in the class of diophantine problems known as specialpoints problems or problems of unlikely intersections . See for instance, [30, 32, 36, 20, 14].The Ax-Schanuel theorem generalizes the Ax-Lindemann-Weierstrass. In the setting of pureShimura varieties, the Ax-Schanuel theorem of [27] has recently been applied to certain dio-phantine problems [15]. Various Ax-Schanuel results have also been applied to various casesof the Zilber-Pink conjecture [3, 11].Over the past several years, in a series of works Aslanyan, and later Aslanyan, Kirbyand Eterovi´c [1, 2, 5, 6, 4] develop the connection between Ax-Schanuel type transcendencestatements and the existential closedness of certain reducts of differentially closed fieldsrelated to equations satisfied by the j -function. This series of work builds on the earlierprogram of Kirby, Zilber and others mainly around the exponential function, see e.g. [22].We expect our results to contribute nontrivially to this line of work. The earlier work onthe exponential function utilizes the transcendence results of Ax [7], where a differentialalgebraic proof of the functional version of Schanuel’s conjecture is given. Intermediatedifferential algebraic results from Ax’s work are utilized in the program studying existentialclosedness results around the exponential function. A different approach was required in [1,2, 5, 6, 4] for studying the j -function, in part since the intermediate results of the functionaltranscendence results of [31] take place in the o-minimal rather than abstract differentialsetting. For instance, the motivation for a differential algebraic proof of Ax-Schanuel resultsand this issue is pointed out specifically following Theorem 1.3 of [5] and in Section 4.4 of [4].Our results open up the possibility of adapting a similar approach to existential closednessaround more general automorphic functions since the general technique of our proof isdifferential algebraic along lines similar to Ax’s work. Besides this issue, it is expected thatour generalizations of the Ax-Schanuel results of [31] can be used to establish existentialclosedness results for more general automorphic functions beyond the modular j -function.1.2. Organization of the paper.
The paper is organized as follows. In Section 2 wegive the necessary background in the Cartan approach to the study of linear differentialequations. In particular, we recall the definition (and basic properties) of a G -principalconnection and its associated Cartan connection form. In Section 3 we prove Theorem Aand derive some of its corollaries. We also introduce the idea of geometric structures (or( G, G/B ) structures) and apply Theorem A in this setting to obtain Corollary B . In Section4 we recall Scanlon’s work on covering maps, show that they are part of the formalism ofgeometric structures and detail the intersection of our work with other similar work inthe literature. In section 5 we use Theorem A to study products of geometric structures(Theorem C) and use those to give a model theoretic study of the (
G, G/B ) structures.Section 6 and 7 are devoted to proving Theorem D, i.e., the full Ax-Schanuel Theorem withderivatives in the case of curves.
DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 4
Acknowledgements.
The main ideas behind this work were developed at the School ofMathematics at the Institute for Advanced Study as a part of the 2019 Summer Collabora-tors Program. We thank the Institute for its generous support and for providing an excellentworking environment. The authors also thank Peter Sarnak for useful conversations duringour time at IAS.2.
The Cartan approach to linear differential equations
In this section, we set up some notations and conventions about principal connections. Acomplete reference is Sharpe’s book [35] or the third part of Epstein-Elzanowski’s book [19].Throughout, we will be working over the field of complex numbers C . Analytic functionsor manifolds mean holomorphic.2.1. Principal connection.
Let G be an algebraic group, Y a smooth algebraic varietyand π : P → Y a principal bundle modeled over G , i.e. endowed with an action of G ,denoted by R or by · , that induces an isomorphism P × G ∼ → P × Y P ( p, g ) ( p, p · g ) = ( p, R g ( p )) . The fibers of π are principal homogeneous G -spaces. The election of a point p in a fiber P y = π − ( y ) induces an isomorphism of G -spaces, G ∼ −→ P y , g p · g, and an isomorphism of groups, G ∼ −→ Aut G ( P y ) , g σ with σ ( p · h ) = p · gh between G and the group Aut G ( P y ) of G -equivariant automorphisms of P y . Note that thispair of isomorphisms conjugate the left action of G on itself with the action of Aut G ( P y ) on P y . A gauge transformation of P is a G -equivariant map F : P → P such that π ◦ F = π .That means that for each fiber F | P y ∈ Aut G ( P y ).We define the vertical bundle T ( P/Y ) as the kernel of dπ , it is a subbundle of T P . A connection is a section ∇ of the exact sequence,0 → T ( P/Y ) → T P → T Y × Y P → . Thus, it is a map ∇ : T Y × Y P → T P, ( v, p )
7→ ∇ v,p , satisfying dπ ( ∇ v,p ) = v .The image ∇ ( T Y × Y P ) ⊂ T P is a distribution of vector fields of rank dim Y on P , theso-called ∇ -horizontal distribution H ∇ . We have a canonical decomposition of the tangentbundle T P = T ( P/Y ) ⊕ H ∇ as the direct sum of its vertical and ∇ -horizontal subbundles. Let D ⊂ T Y be a subset of the tangent bundle of a smooth manifold Y such that for each y ∈ Y, the fiberabove y , D y ⊆ T y Y is a d -dimensional subspace. Further, suppose that for any point y there is an Zariskiopen neighborhood U of y such that for any z ∈ U , we have independent vector fields X ( z ) , . . . , X d ( z )regular vector fields whose span is D z . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 5
A connection induces a ∇ -horizontal lift operator of vector fields on Y to P , ∇ : X Y → π ∗ X P ; v
7→ ∇ v with ( ∇ v )( p ) = ∇ v ( π ( p )) ,p that lifts vector fields in open subset U ⊂ Y up to ∇ -horizontal vector fields in π − ( U ) ⊂ P .This operator is O X -linear. By abuse of notation, this lift operator is represented by thesame symbol ∇ . It completely determines the connection. Note that usually, the symbol ∇ is used to denote the associated covariant derivative. The horizontal lift of rational vectorfields on Y defines a map D : C ( P ) → C ( P ) ⊗ C ( Y ) Ω ( Y ) where Ω ( Y ) is the C ( Y )-vectorspace or rational 1-forms. Such a D extends the differential structure of C ( Y ) given by theexterior derivative, d : C ( Y ) → Ω ( Y ), and satisfies Leibniz rule : D( ab ) = a D( b ) + b D( a ).We say that the connection ∇ is principal if it is G -equivariant: for all ( v, p ) ∈ T Y × Y P and g ∈ G we have ∇ v,R g ( p ) = dR g ( ∇ v,p ). This is equivalent to requiring that the ∇ -horizontal distribution is G -invariant, or that the image of the ∇ -horizontal lift operatorconsists of G -invariant vector fields. We say that the connection ∇ is flat if the lift operatoris a Lie algebra morphism, that is: ∇ [ v,w ] = [ ∇ v , ∇ w ] . In what follows, a connection means a G -principal flat connection. We are mostly concerned with rational connections . The definition above is the definitionof a regular connection. A rational connection on a bundle P over Y is a regular connectionon the restriction P | Y ◦ of the bundle above a Zariski open subset Y ◦ ⊂ Y . Given a G -invariant rational connection ∇ , we may replace the base space Y by a suitable Zariski opensubset Y ◦ such that ∇| Y ◦ is regular. Example 2.1.
The most important example of a rational G -invariant connection is a lineardifferential equation in fundamental form. Let us fix an algebraic irreducible curve Y anda non constant rational function y ∈ C ( Y ). As C ( Y ) is an algebraic extension of C ( y ), thederivation ddy can be uniquely extended to C ( Y ). Our equation is dUdy = A ( y ) U with U an invertible n × n matrix of unknowns and A ∈ gl n ( C ( Y )) . In this situation G is the linear group GL n ( C ) with coordinate ring C h U ji , det ( U ) i , P is Y × G , and the action of G on P is given by right translations: ( y, U ) · g = ( y, U g ). We see ddy as a rational vector field on Y and the linear differential system gives us its ∇ -horizontallift ∇ ddy = ∂∂y + X i,j,k A ji U kj ∂∂U ki . Then ∇ -horizontal lift operator is determined by the above formula and O Y -linearity.The field C ( P ) with the derivation ∇ ddy is called the field of the universal solution of thelinear equation. As the vector fields P k U kj ∂∂U ki are right invariant on G , ∇ is a principal DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 6 connection. The compatibility with Lie bracket is straightforward as for any couple of vectorfields v and w on Y , [ v, w ] is colinear to v ; thus ∇ is flat. The equation defines a regularconnection outside of the set of poles of the rational functions A ji and zeroes of ddy as avector field on Y .2.2. Basic facts about singular foliations.
We will use freely vocabulary and resultsfrom holomorphic foliation theory while recalling the relevant objects in this subsection.A singular foliation F of rank m on an algebraic variety P is a m -dimensional C ( P )-vector subspace of rational vector fields in P , X P , stable by Lie bracket. Such vector fieldsare said to be tangent to F . We say that the foliation F is regular at p ∈ M if there is abasis { v , . . . , v m } of F such that v i ( p ) is defined for i = 1 , . . . , m and v ( p ) , . . . , v r ( p ) are C -linearly independent, otherwise we say that F is singular at p . The set of singular pointsof F form a Zariski closed subset sing( F ) of codimension ≥
2. We say that the foliation F is regular if sing( F ) = ∅ .An integral submanifold of F is an m -dimensional analytic submanifold S ⊂ P (notnecessarily embbeded in P ) whose tangent space at each point is generated by the valuesof vector fields in F . Maximal connected integral submanifolds are called leaves . Throughany regular point passes a unique leaf. Any connected integral submanifold of F determinescompletely a leaf by analytic continuation. For general results about Zariski closures ofleaves of singular foliations we refer to [13], in particular the Zariski closure of a leaf isirreducible.A subvariety V ⊂ P is F -invariant if the vector fields tangents to F whose domain isdense in V restrict to rational vector fields on V . In such case F | V is a singular foliation in V of the same rank. Leaves of F | V are leaves of F contained in V .We say that F is irreducible if and only if it does not admit any rational first integrals,that is, if f ∈ C ( P ) such that vf = 0 for all v ∈ F then f ∈ C . From Theorem 1.4 in [13]we have that F is irreducible if and only if it has a dense leaf. Let L be leaf of F and L its Zariski closure. Then L is an irreducible F -invariant variety and the restricted foliation F | L is irreducible.If ∇ is a rational connection then the spaceΓ rat ( H ∇ ) = C ( P ) ⊗ C ( Y ) ∇ ( X Y )of rational ∇ -horizontal vector field is a singular foliation in P if and only if ∇ is flat.Moreover, if ∇ is a regular flat connection then Γ rat ( H ∇ ) is a regular foliation. By abuse ofnotation Γ rat ( H ∇ )-invariant varieties are called ∇ -invariant varieties and leaves of Γ rat ( H ∇ )are called ∇ -horizontal leaves. If ∇ is regular and L is a ∇ -horizontal leaf then the projection L → Y is a topological cover in the usual topology.Note that the foliation Γ rat ( H ∇ ) has non horizontal leaves (called vertical leaves ) includedin the fibers of P at non regular points of ∇ .2.3. The Galois group.
Let us consider ∇ a regular connection. We say that a variety Z ⊂ P is ∇ -invariant if H ∇ | Z ⊂ T Z and the projection on Y is dominant. The intersectionof ∇ -invariant varieties is ∇ -invariant and therefore for each point p ∈ P there is a minimal ∇ -invariant variety Z such that p ∈ Z . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 7
Frobenius theorem ensures that through any p ∈ P , there exists a unique ∇ -horizontalleaf denoted by L p . It is clear that π | L p : L p → Y is surjective. Moreover if Z ⊂ P is a ∇ -invariant variety and p ∈ Z then the Zariski closure of the leaf, L p ⊂ Z . Note that, bythe G -invariance of the distribution H ∇ , we have L p · g = L p · g . Lemma 2.2.
The following are equivalent: (a) Z is a minimal ∇ -invariant variety. (b) For any p ∈ Z , Z is the Zariski closure of L p . (c) Z is the Zariski closure of a ∇ -horizontal leaf.Proof. ( a ) ⇒ ( b ) Assume that Z is minimal, and let p ∈ Z . As Z is ∇ -invariant it implies L p ⊂ Z and therefore L p ⊂ Z . We have that L p is a ∇ -invariant variety, and then byminimality of Z we have Z = L p .( b ) ⇒ ( c ) Trivial.( c ) ⇒ ( a ) Let us consider p ∈ Z such that Z = L p . Note that Z is irreducible. Let us seethat Z is minimal. Let us consider W ⊂ Z a ∇ -invariant subvariety and q ∈ W . We have, L q ⊆ W ⊆ L p = Z. Let us consider p ′ ∈ L p in the same fiber than q . Note that there is g ∈ G such that q · g = p ′ . Then we have L q · g = Z . If follows that Z and W have the same dimension.They are irreducible and therefore they are equal. (cid:3) Lemma 2.3.
Let Z be a minimal ∇ -invariant variety for a regular connection. Then, Gal( Z ) = { g ∈ G : Z · g = Z } is an algebraic subgroup of G , and π | Z : Z → Y is a Gal( Z ) -principal bundle.Proof. We just need to note that the isomorphism, P × G ∼ −→ P × Y P, ( p, g ) ( p, p · g )maps Z × Gal( Z ) onto Z × Y Z . (cid:3) Note that if Z is a minimal ∇ -invariant variety then any other is of the form Z · g for some g ∈ G and Gal( Z · g ) = g − Gal( Z ) g . It follows that P is the disjoint union of minimal ∇ -invariant varieties, each one of them a principal bundle, all of them modeled over conjugatedsubgroups of G . Definition 2.4.
The
Galois group of ∇ , Gal( ∇ ) is the algebraic group Gal( Z ) for any min-imal ∇ -invariant variety Z . It is a well defined abstract algebraic group, but its immersionas a subgroup of G depends on the choice of Z . Example 2.5.
Let us consider example 2.1. Then, L p is analytic subvariety obtained by theanalytic continuations of the germ of a solution of the linear equation with initial condition p ∈ P along any path in Y starting at π ( p ). The differential field ( C ( L p ) , ∇ ddy ) is a Picard-Vessiot extension of ( C ( Y ) , ddy ). We usually chose a point of the form p = ( y , id) ∈ Y × G so that Galois group is called Picard-Vessiot group at y . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 8
Cartan connection form.
An alternative way to encode a principal connection ∇ isthrough its connection form Ω. First, let us define the structure form ω , which is canonicallyattached to the principal bundle. We differentiate the action R of G on Y , with respect thesecond factor along P × { e } so that we obtain a trivialization d R : P × g ∼ −→ T ( P/Y ) . Such trivialization defines the structure form ω = pr ◦ d R − of the bundle, ω : T ( P/Y ) → g . Note that if g ∈ G and F is a gauge transformation then dR g and dF map the verticalbundle T ( P/Y ) onto itself. Therefore R ∗ g ( ω ) and F ∗ ( ω ) are well defined as g -valued formson T ( P/Y ). The structure form have the following properties:(1) Right G -covariance: for g ∈ G , R ∗ g ω = Adj g − ◦ ω ;(2) Left gauge-invariance: for any gauge transformation F ∗ ( ω ) = ω .(3) For each fiber P y the form ω y = ω | P y satisfy the Maurer-Cartan structure equation: dω y = −
12 [ ω y , ω y ] . Given a principal connection ∇ there is a unique way to extend the structure form ω to a g -valued 1-form Ω on P that vanish along the horizontal distribution, the so-called Cartanconnection form: Ω : T P → g , v p ω ( v p − ∇ dπ ( v p ) ,p ) . It is clear that Ω and ∇ determine each other as H ∇ = ker( ω ∇ ). The gauge-invarianceproperty of ω extends partially to Ω. Definition 2.6.
We say that a gauge transformation F : P → P is a gauge symmetry of ∇ if for any p ∈ P and v ∈ T π ( p ) Y , we have that dF ( ∇ v,p ) = ∇ v,F ( p ) .Summarizing, the connection form Ω, attached to a flat principal connection has thefollowing properties:(1) Ω | T ( P/Y ) = ω .(2) Ω = 0 on the horizontal distribution(3) Ω is G -covariant ; For g ∈ G , g ∗ Ω = Adj g − ◦ Ω (right covariant);(4) Ω is gauge-invariant; for any gauge symmetry F of ∇ , F ∗ Ω = Ω (left invariant)(5) Ω satisfies Cartan structure equation d Ω = −
12 [Ω , Ω] . Example 2.7.
Going on with example 2.1, the structure form ω is, ω = ( U − dU ) | T ( P/Y ) and the connection form is: Ω = U − dU − U − AU dt.
Proposition 2.8.
Let L be a leaf of H ∇ . The restriction of Ω to the Zariski closure of thisleaf, L , takes values in Lie(Gal( L )) . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 9
Proof.
Note that L → Y is a principal Gal( L ) bundle. Therefore, ω | T ( L /Y ) takes values inLie(Gal( L )). If follows that Ω | L takes values in Lie(Gal( L )). (cid:3) ∇ -Special subvarieties and Ax-Schanuel Definition 3.1.
Let ∇ be a flat G -principal connection over Y with Galois group G . Asubvariety X ⊂ Y is ∇ -special if the group Gal( ∇| X ) is a strict subgroup of G .Let H ⊂ G be an algebraic subgroup. A subvariety X ⊂ Y is H -special if Gal( ∇| X ) ⊂ H . Example 3.2.
The connection on the trivial ( C ∗ ) - bundle over C = Y given by dU − (cid:20) d ( y y ) 00 d ( y ) (cid:21) U = 0with solution U = (cid:20) e y y e y (cid:21) has special subvarieties. Its Galois group is C ∗ × C ∗ but itsrestriction to lines y = q ∈ Q has Galois group C ∗ The relation between the ∇ -special subvarieties and special Shimura subvarieties (Hodgetype locus) is not clear from the definitions. Both have a monodromy group and in bothcases the Zariski closure of that monodromy group is not G . Throughout, unless otherwisestated, we use special subvarieties as short for ∇ -special subvarieties.The first version of our Ax-Schanuel theorem is the following. Theorem 3.3.
Let ∇ be a G -principal connection on P → Y with Galois group G . Let V be an algebraic subvariety of P , L is an horizontal leaf and V an irreducible componentof V ∩ L . If dim V < dim( V ) + dim G and dim( V ) > then the projection of V in Y iscontained in a special subvariety. Corollary 3.4.
Let ˆ V : spf C [[ s , . . . , s n ]] → L ⊂ P be a non constant formally parameter-ized space in a horizontal leaf of ∇ and V its Zariski closure. If dim V < rk(ˆ V ) + dim G then the projection of V in Y is a special subvariety.Proof. As V = V ∩ L is a germ of analytic subvariety containing ˆ V then dim V ≥ rk(ˆ V ).The inequality in the hypothesis of the corollary implies the one of the theorem. (cid:3) Proof of Theorem 3.3.
The proof of the theorem follows from the next lemmasabout the connection form Ω. We can assume V is the Zariski closure of the component V of V ∩ L and assume V is irreducible. Lemma 3.5.
The restriction of Ω to V has kernel of positive dimension at the genericpoint.Proof. As dim V > p ∈ V such that the dimension ofthe kernel of Ω | V at p is greater than or equal to dim V is a Zariski closed subset. This setcontains the points of V for which the tangent vectors are in the kernels. This analytic spaceis Zariski dense in V . It hence follows that the rank of Ω | V is smaller than dim V − dim V < dim G . (cid:3) Lemma 3.6. If dim ker(Ω | V ) = dim Y then V = P . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 10
Proof.
As ker(Ω | V ) ⊂ ker(Ω) | V and dim ker(Ω) = dim Y , the horizontal leaf through p ∈ V is included in V . By hypothesis this leaf is Zariski dense in P . (cid:3) Lemma 3.7.
There is a Lie subalgebra k ⊂ g such that ∀ p ∈ V , Ω p ( T p V ) = k .Proof. We show that Ω p ( T p V ) does not depend on p ∈ V . Let e , . . . , e q be a basis of g anddecompose, Ω | V = q X i =1 Ω i e i . We may assume that first Ω , . . . , Ω k form a maximal set of linearly independent 1-formsover C ( V ) among the Ω i ’s. We have then:Ω k + i = k X j =1 b ij Ω j with b ij rational on V . We consider a vector field D on V in the kernel of Ω. By taking Liederivatives we obtain: Lie D (Ω k + i ) = k X j =1 ( D · b ij )Ω j + k X j =1 b ij Lie D Ω j . From Cartan formula (Lie D = i D ◦ d + d ◦ i D ) and Cartan structural equation ( d Ω j is acombination of the 2-forms Ω k ∧ Ω ℓ with constant coefficients) we have that Lie D Ω j = 0 forany j = 1 , . . . , q , and therefore for i = 1 , . . . , q − k we have k X j =1 ( D · b ij )Ω j = 0 . By the linear independence we obtain that D · b ij = 0. It hence follows that the b ij ’s arefirst integrals of the foliation of V defined by ker(Ω | V ) and thus are constant on V . Since V is the Zariski closure of V , the functions b ij are constant. We have thus proved that theimage of Ω p is a fixed linear subspace k ⊂ g .We claim that k is a Lie subalgebra. Indeed, if we let e , e be two elements of k and let v and v be two vector fields on V such that Ω | V ( v i ) = e i for i = 1 ,
2. Then by Cartan’sstructural equation, [ e , e ] = d Ω( v , v )On the other hand, as Ω is a 1-form with values in k , we have that d Ω is a 2-form withvalues in k . (cid:3) Lemma 3.8.
The Lie algebra k is the Lie algebra of an algebraic subgroup K ⊂ G .Proof. There exists a connected Lie subgroup K ⊂ G such that Lie( K ) = k . Choosing apoint p ∈ V ⊂ P makes an identification of the fiber P through p with G such that p isthe identity and the action of G on P is the right translation on G .The image of V = V ∩ P under this identification gives an algebraic variety throughthe identity. As Ω( T V ) = k , its tangent is spanned by the vector fields generating right DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 11 translation by K . So K is the connected component of the identity of V and thus isalgebraic. (cid:3) Consider the quotient of P by the action of K , namely ρ : P → P/K and ℓ : g → g / k . Lemma 3.9.
The foliation ℓ ◦ Ω = 0 is ρ -projectable on a foliation F on P/K of dimensionequal to the dimension of Y . A foliation given by a distibution H on P is ρ -projectable if there is a foliation G on P/K such that H ⊕ ker dρ = dρ − G . This means that the image by ρ of a leaf of H is a leaf of G . Proof.
The distribution given by the kernel of ℓ ◦ Ω is a foliation as the sub Lie algebrastructure of k implies the Frobenius condition. Vector fields v on P with Ω( v ) ∈ k are in thekernel of ℓ ◦ Ω and also in the space ker(Ω). So the foliation has dimension dim Y + dim K and its leaves contain the orbits of K thus it is ρ -projectable. (cid:3) Lemma 3.10.
The dimension of ρ ( V ) is between and dim Y − .Proof. As ℓ ◦ Ω | V = 0, we have that V is included in a leaf of ℓ ◦ Ω. Thus ρ ( V ) is includedin a leaf of F . Such a leaf is a (dim Y )-dimensional immersed analytic subset so thatdim ρ ( V ) ≤ dim Y . If ρ ( V ) is a point then it projects on a point in Y , but this projectionmust contain a non constant formal curve thus 0 < dim ρ ( V ).If ρ ( V ) has dimension dim Y , then ρ ( V ) is an algebraic leaf of F . The existence of sucha leaf implies that the Lie algebra of the Galois group of ∇ is k . This is not possible byhypothesis and then dim ρ ( V ) < dim Y . (cid:3) We can now conclude the proof of Theorem 3.3.
Proof.
As dim ρ ( V ) < dim Y its projection in Y is contained in a strict algebraic subvariety X of Y . It suffices to build a strict ∇| X -invariant subvariety of P | X . But we have that ρ ( V ) is an algebraic subvariety of P/K such that ker dπ | V = 0. So the map ρ ( V ) → X isa finite map. It is an algebraic leaf of F | X this implies that ρ − ( ρ ( V )) is a ∇| X -invariant K -principal subbundle: Gal( ∇| X ) has Lie algebra included in k thus X is special. (cid:3) Uniformizing equation and Ax-Schanuel.
Let G be an algebraic group and B analgebraic subgroup. A ( G, G/B )-structure on an algebraic variety Y is usually defined usingcharts on Y with values in G/B and change of charts in G . Here is a algebraic version ofthis notion.The jet space J ∗ ( Y, G/B ) of invertible jets of map is endowed with an action of G bypostcomposition. As it is a jet space, its ring has a D Y -differential structure. Definition 3.11. A rational ( G, G/B ) -structure on Y is a D Y -subvariety C of J ∗ ( Y, G/B )with a Zariski open Y o ⊂ Y such that C | Y o a G -principal sub-bundle of J ∗ ( Y o , G/B ).As D Y -varieties have no O X torsion, C is well defined if we know C | Y o . Since dim Y =dim G/B , one has an isomorphism from J ∗ ( Y, G/B ) to J ∗ ( G/B, Y ). If C is a ( G, G/B )-structure, then its image under this isomorphism is denoted by U and is a finite dimensional(over C ) D G/B -subvariety of J ∗ ( G/B, Y ). A local analytic solution υ of Y is a holomorphicmap defined on a open subset υ : U → Y , whose jet j ∞ υ : U → J ∗ ( G/B, Y ) takes values in Y . These solutions are called uniformizations of the ( G/B, G )-structure.
DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 12
We call Y the space of uniformizations of the ( G, G/B )-structure C . As the datas C and Y are equivalent, we will use any of them to denote the ( G, G/B )-structure.
Lemma 3.12.
A geometric structure C on an algebraic variety Y is a principal bundlewith a principal connection on some Zariski open subset Y o .Proof. By definition we have that C is a G principal bundle over Y o . The D Y -structureof C gives a lift of vector fields on Y ◦ to C , it is a connection on C | Y ◦ . The group G acts on C by post-composition. The action of D Y is the infinitesimal part of the action bypre-composition. These two commute and hence the connection is G -invariant. (cid:3) The following statement is another (more usual) version of the Ax-Schanuel theorem. Itfollows by applying Corollary 3.4 to C . Corollary 3.13.
Let υ be a uniformization of a ( G, G/B ) -structure on an m -dimensionalalgebraic variety Y . Assume that ˆ γ is a non constant formal curve on G/B such that tr . deg . C C (ˆ γ, ( ∂ α υ )(ˆ γ ) : α ∈ N m ) < G then the Zariski closure υ (ˆ γ ) is a special subvariety of Y From Corollary 3.13 we get the Ax-Schanuel theorem without derivatives as stated forinstance in [27].
Corollary 3.14.
Let υ be a uniformization of an irreducible ( G, G/B ) -structure on an m -dimensional algebraic variety Y . Assume W ⊂ G/B × Y is an irreducible algebraicsubvariety intersecting the graph of υ . Let U be an irreducible component of this intersectionsuch that dim W < dim U + dim Y. Then the projection of U to Y is contained in a special subvariety of Y .Proof. Let j : C → G/B × Y be the 0-jet projection. Then ( j ) − ( W ) intersect the graphof the jet of υ on ˜ U such that j ( ˜ U ) = U .Now dim( j − ( W )) = dim W + dim B < dim U + dim X + dim B = dim ˜ U + dim G . UsingCorollary 3.13 the result follows. (cid:3) We also obtain the following.
Corollary 3.15.
Let υ be a uniformization of a ( G, G/B ) -structure on a m dimensionalalgebraic variety Y . Let A ⊂ G/B be an irreducible algebraic subvariety. If υ ( A ) has aproper Zariski closure in Y , then υ ( A ) is contained in a special subvariety of Y .Proof. Let U ⊂ G/B × Y be the graph of υ | A and V be its Zariski closure. Thendim V ≤ dim A + dim υ ( A ) < dim U + dim X. (cid:3) DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 13 Differential equations and covering maps
Our main technical results have applications to algebraicity problems for certain analyticfunctions, where our applications overlap with some existing results. In this section, weoutline some settings for existing results which overlap our applications. We work in thecontext of covering maps following [34], but we will explain the setup.Let G be an algebraic group over C with a regular action on a complex variety X suchthat X = G/B for some algebraic subgroup B of G . Let U be a complex submanifold of X ( C ) and Γ be a Zariski dense subgroup of G ( C ) with the property that the induced actionof Γ on X preserves U . Assume that we have a complex analytic map υ : U → Y which isa covering map of the complex algebraic variety Y expressing Y ( C ) as Γ \ U . So, υ − is amultivalued function with branches corresponding to elements of Γ.Under those assumptions Scanlon shows that there is a differential algebraically con-structible function ˜ χ : X → Z , for some algebraic variety Z , called the generalized Schwarzianderivative associated to υ such that for any differential field F having field of constants C and points a, b ∈ X ( F ) one has that ˜ χ ( a ) = ˜ χ ( b ) if and only if a = gb for some g ∈ G ( C ).For the situation we have in mind, the restriction to differential fields F with m := dim X commuting derivatives is enough. From Scanlon’s construction, it follows that the general-ized Schwarzian derivative is defined on an order k jet space ˜ χ : J ∗ k,m ( X ) → Z . Thus themap χ := ˜ χ ◦ υ − : J ∗ k,m ( Y ) → Z is a well-defined analytic map and induces a differentialanalytic map Y ( M ) → Y ( M ) for M any field of meromorphic functions in m variables.Now, also assume that the restriction of υ to some set containing a fundamental domainis definable in an o-minimal expansion of the reals as an ordered field. One of the mainresults of [34], Theorem 3.12, is that under this assumption the function χ := ˜ χ ◦ υ − , forany choice of a branch of υ − , is also differential algebraically constructible. The function χ is called the generalized logarithmic derivative associated to υ . Example 4.1.
When G = PSL , it is classically known that ˜ χ : J ∗ ( X ) → C can be takento be the Schwarzian derivative S ( x ) = (cid:16) x ′′ x ′ (cid:17) ′ − (cid:16) x ′′ x ′ (cid:17) and χ : J ∗ ( Y ) → C is given by S ( y ) + R ( y ) y ′ with R ( y ) = S ( υ − ).The definability property in Scanlon’s theorem is analogous to the hypothesis that canbe made on the principal connection S ( x ) = R ( y ) to have all singulatities regular. Bothimply that R is a rational function.Using χ and ˜ χ , we can define the differential equation satisfied by υ and υ − . Fromthe composition of jets one gets two maps: c : J ∗ k,m ( X ) × J ∗ k,m ( Y ) → J ∗ k ( X, Y ) and c : J ∗ k,m ( Y ) × J ∗ k,m ( X ) → J ∗ k ( Y, X ). The algebraic subvariety given by χ − ¯ χ = 0 projectsby c on a algebraic subvariety of J ∗ k ( X, Y ) and by c on a algebraic subvariety of J ∗ k ( Y, X )Let us assume that ¯ t = ( t , . . . t m ) are the coordinates on U ⊂ X = G/B for somealgebraic subgroup B of G and ¯ y = ( y , . . . y m ) be coordinates on Y . By construction, wehave that υ (¯ t ) satisfies the algebraic differential equation Y = c ( { χ (¯ y ) − ˜ χ (¯ t ) = 0 } ) ⊂ J ∗ k ( G/B, Y )and the inverse branches υ − (¯ y ) satisfies C = c ( { χ (¯ y ) − ˜ χ (¯ t ) = 0 } ) ⊂ J ∗ k ( Y, G/B ) . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 14
Remark . The ∗ appearing in the jet space J ∗ k means that we are adding to the explicitequations above inequations ensuring that the rank of the jacobian matrix of solutions is m . Proposition 4.3.
The equations above define a rational ( G, G/B ) -structure C on Y .Proof. Since ¯ y are coordinates on Y , using c it follows that χ (¯ y ) is a rational function R on Y . Let Y ◦ be the domain of R on which our differential equation can be written ¯ χ (¯ t ) = R ( y )in J ∗ k ( Y, G/B ). By construction, it follows that C coincides with the differential subvarietyof J ∗ ( Y, G/B ) defined by this equation.By Scanlon’s construction, we have a map C × G → C insuring that C is a G -principalbundle on Y ◦ . As a smooth finite dimensional D Y ◦ -subspace of J ∗ ( Y ◦ , G/B ), C gives riseto a connection. The properties of ¯ χ ensure that this connection is G -invariant. (cid:3) It hence follows that our Ax-Schanuel Theorems (corollaries 3.13 and 3.14) hold in thecase of covering maps given in the Scanlon theory. In the coming subsections, we willdescribe the settings in which existing versions of an Ax-Schanuel Theorem for coveringmaps exists and have some overlap with our results.4.1.
Modular curves.
Let j : H → A ( C ) be the classical modular j -function. In thenotation above we take G = SL and B = (cid:26)(cid:18) a c d (cid:19) : ad = 1 (cid:27) the subgroup of lower triangular matrices so that X = SL ( C ) /B ∼ = CP . As well-known,if we take Γ = SL ( Z ), then the quotient Y (1) = Γ \ H can be identified with the affineline A ( C ). We take U to be the open subset of H such that j : U → A ( C ) \ { , } is acovering map. The restriction of j to the domain F = { z ∈ H : | Re ( z ) | ≤ and Im ( z ) ≥ √ } , which contain a fundamental domain, is definable in R an,exp . Hence j is a solution toa ( G, G/B )-structure as describe above. In this case, this structure can be taken to be thewell-known Schwarzian differential equation satisfied by j .In this setting, the Ax-Schanuel Theorem is a result of Pila and Tsimerman [31, Theorem1.1] Theorem 4.4.
Let V ⊂ ( P ) n × Y (1) n be an algebraic subvariety, and let U be a componentof V ∩ Γ( j n ) , where Γ( j n ) denotes the graph of the j -function applied to H inside each of the n copies of P . Then dim U = dim V − n unless the projection of U to Y (1) n is containedin a proper weakly special subvareity of Y (1) n . Replacing Γ in the above results with Γ( N ) , the kernel of the reduction mod N mapSL ( Z ) → SL ( Z /N Z ) (also replacing j with suitable j N and Y (1) with Y ( N )), one obtainsthe same result; j and j N are interalgebraic as functions over C . Pila and Tsimerman [31,Theorems 1.2 and 1.3] prove the more general version of Ax-Schanuel which includes thefirst and second derivatives of j (and replaces n with 3 n ). Our work gives a uniform proofof this result for Γ ⊂ SL ( R ) any Fuchsian group of the first kind. DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 15
Pure Shimura Varieties.
We follow closely the exposition given in [9]. Let G to bea connected semi-simple algebraic Q -group and K a maximal compact subgroup of G ( R ).Then it follows that Ω = G ( R ) /K is a bounded symmetric domain. It is known (cf. [21,Proposition 7.14]) that the compact dual ˇΩ of Ω is given as the quotient ˇΩ = G ( C ) /B fora Borel subgroup B and is a homogeneous projective variety. One can always assume that K ⊂ B , so that Ω is a semi-algebraic subset of ˇΩ.Given an arithmetic lattice Γ ⊂ G ( Q ), the analytic quotient Y := Γ \ Ω = Γ \ G ( R ) /K has the structure of an algebraic variety and is called a pure (connected) Shimura variety.The quotient map q : Ω → Y := Γ \ Ω is a covering map and the result [23, Theorem 1.9]shows that it is definable in R an,exp on some fundamental domain. Hence q is a solution toa ( G, G/B )-structure on Y as defined above.We fix Y = Γ \ Ω a connected pure Shimura variety and q : Ω → Y the quotient map. Definition 4.5. A weakly special subvariety of Y is a Shimura variety Y ′ given as Y ′ = Γ ′ \ G ′ ( R ) /K ′ where G ′ is an algebraic Q -subgroup of G , the group Γ ′ = Γ ∩ G ′ ( Q ) is an arithmetic lattice,and K ′ = K ∩ G ′ ( R ).It follows that if Y ′ is a weakly special subvariety of Y , then Y ′ is algebraic, which bydefinition means that there is an algebraic subvariety V of ˇΩ such that Y ′ = V ∩ Ω. Theorem 4.6. [37, Theorem 1.2]
An irreducible subvariety Z ⊂ Y is weakly special if andonly if some (all) components of q − ( Z ) are algebraic. An essential part of Pila’s strategy for attacking various diophantine problems associatedwith the geometry of certain analytic covering maps (see e.g. [29]) is to identify the (weakly)special subvarieties of Y . In this setting, the most general transcendence result to whichour applications are related comes from [27]. Theorem 4.7. [27, Theorem 1.1]
Let W ⊂ Ω × Y be an algebraic subvariety. Let W be acomponent of W ∩ D of positive dimension, where D is a the graph of the map q : Ω → Y. Suppose that dim
W < dim W + dim Y. Then the projection of W to Y is contained in a proper weakly special subvariety of Y . Ball quotients.
In this subsection we outline the setting of the recent manuscript[10]. Let G := PU( n, B n ⊂ C n . Concretely, let U( n,
1) denote the group of linear transformations of C n +1 leaving invariantthe form: z ¯ z + z ¯ z + . . . + z n ¯ z n − z ¯ z , namely, U( n,
1) = (cid:26) g ∈ GL n +1 ( C ) | g T (cid:18) I n − (cid:19) ¯ g = (cid:18) I n − (cid:19)(cid:27) . To obatain a covering map, one might need to restrict q to an open subset U of Ω avoiding ramificationpoints of the original map. DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 16
For g ∈ U( n, , define the map φ g : B n → B n as follows: if g = (cid:18) A a a a (cid:19) where A is a n × n matrix, a is a column vector, and a is a row vector, then for z ∈ B n ,φ g ( z ) = Az + a a z + a . It is not hard to show that the map φ g is the identity if and only if g ∈ { e it I n +1 : t ∈ R } ∼ = S . Then we define PU( n,
1) = U( n, / S and note that PU( n,
1) is the group ofholomorphic automorphisms of B n . Using [21, Proposition 7.14] it follows that the compact dual, X = CP n , of B n can bewritten as a quotient G/B for some algebraic subgroup B of G . If we let Γ ⊂ G be alattice and let Y be the quotient Γ \ B n , then the quotient map υ : B n → Y is a coveringmap. From [10, Theorem 3.4.5], we have that υ is definable in R an,exp on some fundamentaldomain. Hence, once again, the uniformizer υ is a solution to a ( G, G/B )-structure on Y as defined above. Note that by results of Mok [25], the quotient Y has the structure ofa quasi-projective algebraic variety. In this setting, Baldi and Ullmo [10, Theorem 1.22]established the Ax-Schanuel conjecture: Theorem 4.8.
Let W ⊂ B n × Y and Π be the graph of the quotient map. Let U be anirreducible component of W ∩ Π such that codim U < codim W + codim Π or equivalently dim W < dim U + dim Y. If the projection from U to Y is positive dimensional, then it iscontained in a strict totally geodesic subvariety of Y . Theorem 4.8 generalizes the earlier non-arithmetic Ax-Lindemann-Weierstrass theoremof Mok [26]. In the setting of this subsection, by Corollary 5.6.2 of [10], the totally geodesicsubvarieties are precisely the bi-algebraic subvarieties for the map υ. Keeping in mind thisconnection will be essential later for observing applications of our results which generalizethe Ax-Schanuel result of [10]. There are also similar results of [8] (stated in terms ofgeodesic subvarieties) for Γ ≤ SO ( n, . 5. The product case and applications to Model theory
In this section we use Theorem 3.3 to study products of (
G, G/B )-structures. In par-ticular we show that an Ax-Schanuel type theorem holds in this setting. We then applythis result to give a model theoretic study of set defined, in a differentially closed field, bythe (
G, G/B ) structures. We show that the definable sets are strongly minimal, geometri-cally trivial and in the case of covering maps, satisfy a weak form of the Ax-Lindemann-Weierstrass Theorem with derivatives. One would simply have to establish that the uniformizers are o-minimally definable.
DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 17
Product of ( G, G/B ) -structures. In this subsection we apply some of the results ofSection 3 to products of (
G, G/B )-structures. We first observe that the following Lemmaholds.
Lemma 5.1.
Let G be an algebraic group, B a subgroup and Y ⊂ J ∗ ( G/B, Y ) a rational ( G, G/B ) -structure on Y . If the Galois group of the associated charts set C is G then (1) Y is irreducible, (2) there is no proper D G/B - subvariety of Y .Proof. Using the isomorphism between J ∗ ( G/B, Y ) and J ∗ ( Y, G/B ), we need to prove that C is irreducible. The hypothesis on the Galois group means that C is the Zariski closure ofan horizontal leaf. Irreducibility of the Zariski closure of a leaf of an holomorphic foliationis proved in [13].The isomorphism between J ∗ ( G/B, Y ) and J ∗ ( Y, G/B ) exchanges the two differentialstructures thus a proper D G/B -subvariety of Y gives a proper D Y -subvariety of C . By ourGalois assumption there is no D Y -subvariety of C . (cid:3) For i = 1 . . . n let G i be simple algebraic groups with trivial center, B i ⊂ G i a subgroupand C i ⊂ J ∗ ( Y i , G i /B i ) a rational ( G i , G i /B i )-structure on Y i whose G i -invariant connectionis denoted by ∇ i . As n Y i =1 J ∗ ( Y i , G i /B i ) ⊂ J ∗ n Y i =1 Y i , n Y i =1 G i /B i ! = ˜ J the subset C = Q i C i is a ( Q i G i , Q i G i /B i )-structure on Y = Q i Y i whose connection willbe ∇ . A uniformization of this structure is a map ¯ υ = ( υ , . . . , υ n ), where the i th factor isa uniformization of Y i and depends only of variables G i /B i . Theorem 5.2.
Let V be an algebraic subvariety of ˜ J and ¯ υ a unifomization with graph L in ˜ J . Assume (1) The Galois group of the i th factor is G i , (2) V is the Zariski closure of V ∩ L and dim( V ∩ L ) > , (3) dim V < dim( V ∩ L ) + dim G + . . . + dim G n , (4) the projection of V on each G i /B i is dominant.Then (a) there exist two indices i < j such that the projection X ij of V in Y i × Y j is properspecial subvariety whose projections on both factors is onto. (b) Denote by π i and π j the projections of X ij on Y i and Y j . There is an isomorphism ϕ : π ∗ i C i → π ∗ j C j defined over V ij such that ϕ ∗ π ∗ i ∇ i = π ∗ j ∇ j . (c) Assume the factors have finitely many maximal irreducible { e } -special subvarietiesthen X ij is a correspondence.Proof. (a) Let X be the projection of V in Y × . . . × Y n . By assumption (3) and Theorem3.3, X is a special subvariety of Y × . . . × Y n . The projection C to C i is compatible with DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 18 the connections. As the projection of V on G i /B i is dominant so is the projection of V on Y i and then of X on Y i . This implies that the projection from C | X in C i has a ∇ i -invariantimage. By lemma 5.1, this projection is dominant. If follows that Gal( ∇| X ) is a subgroupof G × . . . × G n that projects onto each component G i . Then, by Goursat-Kolchin lemmathere are indices i < j and an algebraic group isomorphism, σ : G i → G j such that Gal( ∇| X ) ⊂ ( ( g , . . . , g n ) ⊂ Y i G i : σ ( g i ) = g j ) Now, let us consider ∇ ij = ∇ i × ∇ j as a connection on C i × C j → Y i × Y j . It follows, Gal( ∇ ij | V ij ) ⊂ G σ = { ( g , g j ) ∈ G i × G j : σ (¯ g i ) = ¯ g j } . Therefore X ij ⊂ Y i × Y j isa proper special subvariety.(b) Let T ij be the Zariski closure of an horizontal leaf of ∇ ij | X ij with Galois group G σ then T ij ⊂ C i × C j is the is the graph of the connection preserving isomorphism ϕ of thestatement.(c) We may consider Y ◦ i ⊂ Y i the complement of { e } -special subvarieties. One can restrictbundles and connections above the products of Y ◦ i . Thus we may assume that factors haveno { e } -special subvarieties.As X ij ⊂ V i × V j is a G σ -special subvariety and for y ∈ Y i , { y } × Y j is a ( { e } × G )-specialsubvariety, then the intersection X j = X ij ∩ { y } × Y j is empty, has dimension 0 or is aspecial subvariety with group G σ ∩ ( { e } × G ). As G σ is the graph of an isomorphism thelatter is { e } × { e } . As Y j has no { e } -special subvarieties, X j is empty of has dimension 0.The projections of X ij are onto thus X j is generically (on y ) not empty. Then dim X ij =dim Y i and dim X ij = dim Y j and the two projections are dominant. This proves the asser-tion (c), that is X ij is a correspondence. (cid:3) In the following definition, we extract some of the key properties of geometric structuresneeded to apply the above theorem and to formulate a number of applications
Definition 5.3.
A (
G, G/B ) structure Y (or C ) on an algebraic variety Y is said to be simple if(1) G is a centerless simple group,(2) The Galois group of C is G (3) Y as finitely many maximal irreductible { e } -special subvarieties. Remark . If dim Y = 1 the condition (3) trivially holds since any curve with a geometricstructure has no special subvarieties. Example 5.5.
Simple ball quotients are described by Appell’s bivariate hypergeometricsystems F . It is a rank 3 linear connection on a vector bundle on Y = CP × CP with sin-gularities along 7 lines: { , , ∞} × CP , CP × { , , ∞} and the diagonal. This connectiongives a rational (PSL ( C ) , CP )-structure on Y called C hyp or Y hyp (see [38]). DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 19
For clever choices of the exponents in F system, see for instance [17], solutions of Y hyp are built from the quotient of the ball B ⊂ CP by a lattice Γ ⊂ PSU(2 , ⊂ PSL ( C ). Asthis lattice is included in Gal( C hyp ), the Galois group of this geometric structure is PSL ( C )and it satisfies (1) and (2) in definition 5.3.To see that the third condition is satisfied, let us consider X a { e } -special curve in Y and denote υ : B → Y the quotient map. The condition Gal( C hyp | X ) = { e } implies that therestriction of an inverse branch υ − | X is a rational map on X with values in B . As X iscomplete, by Liouville theorem υ − | X must be constant which is in contradiction with thedefinition of C hyp . Corollary 5.6.
Let ( Y, Y ) be a simple ( G, G/B ) -structure on Y , ˆ t . . . ˆ t n be n formalparametrizations of (formal) neighborhoods of points p , . . . p n in G/B and υ , . . . , υ n besolutions of Y defined in a neighborhood of p , . . . p n respectively. If tr . deg . C C (cid:16) ˆ t i , ( ∂ α υ i )(ˆ t i ) : 1 ≤ i ≤ n, α ∈ N dim Y (cid:17) < dim Y + n dim G then there exist i < j such that tr . deg . C C ( υ i (ˆ t i ) , υ j (ˆ t j )) = tr . deg . C C ( υ i (ˆ t i )) = tr . deg . C C ( υ j (ˆ t j )) = dim G. Strong minimality of the differential equations for uniformizers.
We beginby recalling some of the relevant notions from the model theoretic approach to the studyof differential equations. We let L m = { , , + , ·} ∪ ∆ denote the language of differentialrings, where ∆ = { ∂ , . . . , ∂ m } is a set of unary function symbols. From a model theo-retic perspective, differential fields are regarded as L m -structures where the symbols ∂ i areinterpreted as derivations, while the other symbols interpreted as the usual field operations.A differential field ( K, ∆) is differentially closed if it is existentially closed in the sense ofmodel theory, namely if every finite system of ∆-polynomial equations with a solution in a∆-field extension already has a solution in K . We use m- DCF to denote the common firstorder theory of differentially closed fields in L m . It follows that m- DCF has quantifierelimination, meaning that every definable subset of a differentially closed field ( K, ∆) is aboolean combination of Kolchin closed sets. Remark . We will use the following model theoretic conventions(1) The notation υ will be used both for a tuple and an element.(2) We say that a tuple υ is algebraic over a differential field K , and write υ ∈ K alg , ifeach coordinate of υ is algebraic over K .We fix a saturated model ( U , ∆) of m- DCF and assume that C , the field of complexnumbers, is its field of constants of, i.e., C = { υ ∈ U : ∂ ( υ ) = 0 for all ∂ ∈ ∆ } . Given adifferential field subfield K of U and υ a tuple of elements from U , the complete type of υ over K , denoted tp ( υ/K ), is the set of all L m -formulas with parameters from K that υ satisfies. It is not hard to see that the set I p,K = { f ∈ K { X } : f ( X ) = 0 ∈ p } = { f ∈ K { X } : f ( υ ) = 0 } The description given here is not a first order axiomatization. We refer the reader to [24] for the basicmodel theory of m-
DCF . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 20 is a differential prime ideal in the differential polynomial ring K { X } , where p = tp ( υ/K ).Using quantifier elimination, it is not hard to see that the map p I p,K is a bijection be-tween the set of complete types over K and differential prime ideals in K { X } . Furthermore,it follows that a tuple υ is a realization of tp ( υ/K ) if and only if I p,K is the vanishing idealof υ over K . Therefore in what follows there is no harm to think of p = tp ( υ/K ) as theideal I p,K Definition 5.8.
Let K ⊂ U be a differential field and υ is a tuple from U .(1) Let F ⊂ U be a differential field extension of K . We say that tp ( υ/F ) is a non-forking extension of tp ( υ/K ) if K h υ i is algebraically disjoint from F over K , i.e, if y , . . . , y k ∈ K h υ i are algebraically independent over K then they are algebraicallyindependent over F .(2) We say that tp ( υ/K ) has U -rank 1 (or is minimal) if and only if υ K alg but everyforking extension of tp ( υ/K ) is algebraic, that is has only finitely many realizations. Remark . Let K and υ be as above and let p = tp ( υ/K ). Let F be a differential fieldextension. Assume further that tr.deg. K K h υ i = r .(1) If K is algebraically closed then p has a unique non-forking extension to F , namely tp (ˆ υ/F ) for any ˆ υ realizing p such that tr . deg . F F h ˆ υ i = r .(2) We have that tp ( υ/F ) is algebraic if and only if υ ∈ F alg .(3) Also tp ( υ/F ) is a nonforking extension of tp ( υ/K ) if and only if tr . deg . K K h υ i =tr . deg . F F h υ i .(4) In particular, the assumptions p has U -rank 1 and tr . deg . F F h υ i < r (so that tp ( υ/F ) is a forking extension of p ) implies that υ ∈ F alg .Let Y ⊂ U ℓ be a definable set and K any differential field over which Y is defined.Assume that the order of Y , ord( Y ) = sup { tr . deg . K K h υ i : υ ∈ Y } , is finite; say ord( Y ) = r . By the (complete) type p of Y over K we mean that p = tp ( υ/K ) for any υ ∈ Y such thattr . deg . K K h υ i = r . Recall that we say that Y is strongly minimal if it cannot be writtenas the disjoint union of definable sets of order r , and for any differential field extension F of K and element υ ∈ Y , we have that tr.deg ( F h υ i /F ) = 0 or r . We will make use of thefollowing fact. Fact 5.10.
The definable set Y is strongly minimal if and only if its type over K has U-rank 1, if Y cannot be written as the disjoint union of K -definable sets of order r , and for any element υ ∈ Y , we have that tr.deg ( K h υ i /K ) = 0 or r . Definition 5.11.
Let Y ⊂ U m be a strongly minimal set and K any differential field overwhich Y is defined. We say that Y geometrically trivial if for any distinct υ , . . . , υ ℓ ∈ Y , ifthe collection consisting of υ , . . . , υ ℓ together with all their derivatives ∂ α υ i is algebraicallydependent over K then for some i < j , the pair υ i , υ j together with their derivatives arealgebraically dependent over K . This formulation is precisely suited for the argument we give later in the paper, and we have phrasedit in this way so that it might be used more easily as a black box for non-experts. Taken together, theconditions might equivalently be written as - Y is the zero set of a prime differential ideal P , such that anydifferential ideals containing P (even after base change to a larger differential field) have the property thattheir zero sets are finite. DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 21
We now aim to show that the set defined by the partial differential equations for any uni-formizer is strongly minimal and geometrically trivial. We will need a basic tool from modeltheory (more precisely stability theory) sometimes called the
Shelah reflection principle . Werestrict our exposition to types of finite order. Let F = F alg be any algebraically closeddifferential field and let p = tp ( υ/F ) for some tuple υ . Assume that tr . deg . F F h υ i = ℓ .We say that a sequence ( υ i ) ∞ i =1 is a Morley sequence in p if υ i +1 realizes the (unique) nonforking extension of p over F i = F h υ , . . . , υ i i alg , i.e., in particular tr . deg . F i F i h ¯ υ i +1 i = ℓ .It follows that one can take υ = υ .In general, when given a differential variety, Y defined over a differential field K , p thetype of a generic solution of Y over K , υ a realization of p , and F a differential field extensionof K , we say that tp ( υ/F ) is a forking extension of p if tr . deg . F F h υ i < tr . deg . K K h υ i . Otherwise, tp ( υ/F ) is a non forking extension of p (if K is algebraically closed, then thisextension is unique). The next result gives a characterization of the kinds of fields one needsto consider while characterizing forking extensions of a type. The Shelah reflection principle. [33, Lemma 2.28]
Let K be any differential field andlet p = tp ( υ/K ) for some tuple υ with tr . deg . K K h υ i = r . Let F = F alg be an algebraicallyclosed differential field extension such that tp ( υ/F ) is a forking extension of p . Then thereis a finite initial segment ( υ , . . . , υ k ) of a Morley sequence ( υ i ) ∞ i =1 in tp ( υ/F ) such that tr . deg . K K h υ , . . . , υ k i < k · r . The intuition here is that we start with a Morley sequence in tp ( υ/F ) and forking iscaptured in the fact that at some point this appropriately chosen sequence ceases to be aMorley sequence in p = tp ( υ/K ) (recall that we can start with υ = υ ). We can now applyTheorem 5.2 to study the set defined by the differential equations for uniformizers. Let usfirst explain the translation from the geometrical setting to m- DCF .We have fixed a ( G, G/B )-structure C on an algebraic variety Y with the group G simple and centerless and with dim( G ) = k . We assume that the Galois group of theassociated charts connection on C is G . We consider an open subset of U of G/B andassume that ¯ t = ( t , . . . t m ) are the coordinates on U realizing a transcendence basis of C ( G/B ). We assume Y ⊂ C ℓ . Now a uniformization of the ( G, G/B )-structure on Y , say υ : U → Y ⊂ C ℓ , will be described by a system of partial differential equations in variables t , . . . t m and unknowns the ℓ coordinates of υ .We assume throughout that our universal field U contains elements t , . . . t m such that ∂ i t i = 1 and ∂ j t i = 0. We denote by Y the ( ∂ , . . . , ∂ m )-differential equations satisfied by υ together with the inequations ensuring that the rank of the jacobian matrix of solutionsis m . By abuse of notation, Y ⊂ U ℓ also denotes the solution set it defines. Next let K ,with C ⊆ K ⊆ C (¯ t ) alg , for some (any) differential field over which Y is defined. In general K = C . Remark . We have the following observations Note that this implies (assuming one selects a sequence with the given properties of minimal length) thatwe have tr . deg . F F h υ k i ≤ tr . deg . K h υ ,...,υ k − i K h υ k i < r , while ( υ , . . . , υ k − ) is a Morley sequence over K .In fact, one can actually arrange that tr . deg . F F h υ k i = tr . deg . K h υ ,...,υ k − i K h υ k i or even more specificallythat a canonical base for the forking extension is contained in the algebraic closure of the initial segment ofthe Morley sequence. DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 22 (1) The assumption on the rank of the jacobian matrix is implicitly part of the formalismof Subsection 3.2. When dim ( Y ) = 1, this corresponds to the assumption that oneonly considers non-constant solutions.(2) Using Lemma 5.1 (2) for any υ ∈ Y , we have that tr.deg. C (¯ t ) C (¯ t ) h υ i = k . Hence italso follows that tr.deg. K K h υ i = k .We need the following abstract reformulation of Corollary 5.6. Corollary 5.13.
Let υ , . . . , υ n ∈ Y be distinct solutions. Iftr.deg. C (¯ t ) C (¯ t ) h υ , . . . , υ n i < kn, then for some i < j , we have that tr.deg. C ( υ i ) C ( υ i , υ j ) ≤ m − . If we further assume that ( Y, Y ) has finitely many maximal irreducible { e } -special subvari-eties then υ i ∈ C ( υ j ) alg , that is each component of υ i is algebraic over C ( υ j ) .Proof. Assume that tr.deg. C (¯ t ) C (¯ t ) h υ , . . . , υ n i = r < kn and let C be a finitely generated(over Q ) algebraic closed subfield of C such thattr.deg. C ( t ) C (¯ t ) h υ , . . . , υ n i = r. For example we can take C to be generated by the coefficients of the polynomial definingthe algebraic relations over C (¯ t ) between υ , . . . , υ n and derivatives.Applying Seidenberg’s embedding theorem to the field C (¯ t ) h υ , . . . , υ n i , we may assumethat υ , . . . , υ n are elements of M ( U ), the field of meromorphic functions on an open con-nected domain U ⊂ C m . Using Theorem 5.2, sincetr.deg. C C (¯ t ) h υ , . . . , υ l +1 i < kn + m, for some 1 ≤ i ≤ n , we have that tr.deg. C ( υ i ) C ( υ i , υ j ) ≤ m − Y, Y ) has finitely many maximal irreducible { e } -special sub-varieties then from Theorem 5.2 (c) we get that υ i ∈ C ( υ j ) alg . (cid:3) Theorem 5.14.
Assume that ( Y, Y ) is simple. Then Y is strongly minimal and geometri-cally trivial. Furthermore, if we let ( F, ∂ , . . . , ∂ m ) be a differential extension of K and let υ , υ ∈ Y with υ , υ Y ( F alg ) then if υ ∈ F h υ i alg , we have that υ ∈ C ( υ ) alg .Proof. Using Lemma 5.1 and Fact 5.10 (see also remark 5.12) to show that Y is stronglyminimal, all we have to show is that its type over K has U -rank 1. Let υ ∈ Y be suchthat p = tp ( υ/K ) is the type of Y over K . As pointed out in Remark 5.12 we havethat tr.deg. K K h υ i = k . We need to show that every forking extension of p is algebraic.Suppose that F = F alg is a differential field extension of K such that q = tp ( υ/F ) is aforking extension of p . Using the Shelah reflection principle we can hence find distinct υ , . . . , υ r +1 ∈ Y , an initial segment in a Morley sequence in q , such that • tr.deg. K K h υ , . . . , υ r i = k · r ; but • tr.deg. K K h υ , . . . , υ r +1 i < k · ( r + 1). DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 23
Using Corollary 5.13, since tr.deg. C (¯ t ) C (¯ t ) h υ , . . . , υ r +1 i < k · ( r + 1) , for some 1 ≤ i ≤ r ,we have that υ r +1 ∈ C ( υ i ) alg . Note that we use here that υ , . . . , υ r , and derivatives arealgebraically independent over K . Hence tr.deg. K K h υ , . . . , υ r +1 i = k · r , that is υ r +1 ∈ K h υ , . . . , υ r i alg . It hence follows that the only way forking can occur is if q is algebraicwhich is what we aimed to show.Finally, given that Y is strongly minimal, the statement of Corollary 5.13 is preciselygeometric triviality. The last implication is a direct consequence of Corollary 5.13 andgeometric triviality. (cid:3) The case of covering maps.
Assume now that υ : U → Y is a covering map. So wehave Γ a Zariski dense subgroup of G ( C ) with the property that the induced action of Γon G/B preserves U and υ is a covering map of the complex algebraic variety Y expressing Y ( C ) as Γ \ U . We also assume that the restriction of υ to some set containing a fundamentaldomain is definable in an o-minimal expansion of the reals as an ordered field.Let Comm G (Γ) be the commensurator of Γ. Recall that by a Comm G (Γ)-correspondence(also known as Hecke correspondence) on Y ( C ) × Y ( C ) we mean a subset of the form X g = { υ (¯ τ ) × υ ( g · ¯ τ ) : ¯ τ ∈ U } where g ∈ Comm G (Γ). It follows that X g is given by equations Φ g ( X, Y ) = 0 for someset Φ g of polynomials with complex coefficients. So Φ g ( υ (¯ t ) , υ ( g ¯ t )) = 0. With this nota-tion, for g , g ∈ G ( C ) we more generally say that υ ( g ¯ t ) and υ ( g ¯ t ) are in Comm G (Γ)-correspondence if Φ g ( υ ( g ¯ t ) , υ ( g ¯ t )) = 0 for some g ∈ Comm G (Γ). In other words if g and g are in the same coset of Comm G (Γ). By analogy with the case of curves, the set ofpolynomials Φ g is called Γ-special.We keep the assumptions and notations from the previous subsection: we assume thatour universal differential field U contains elements t , . . . t m such that ∂ i t i = 1 and ∂ j t i = 0.By abuse of notation, we denote by Y the set of solution of the ( G, G/B ) structure for υ and we write K , with C ⊆ K ⊆ C (¯ t ) alg , for some (any) finitely generated differential fieldextension of C over which Y is defined. We also write k = dim G . Proposition 5.15.
Let υ , υ ∈ Y be two distinct solutions. There is an embeddingof K h υ , υ i into the field of meromorphic functions on some open connected domain V contained in the fundamental domain of Γ such that υ i = υ ( g i ¯ t ) for some g i ∈ G ( C ) .Consequently, if there is a set P of polynomials in C [ X, Y ] such that P ( υ , υ ) = 0 (ie υ ∈ C ( υ ) alg ), then P is Γ -special.Proof. We first show that we can write υ i = υ ( g i ¯ t ) for some g i ∈ G ( C ), where υ : U → Y isthe covering map. Let V ⊂ U be a open connected domain which is properly contained ina fundamental domain of action of Γ. Applying Seidenberg’s embedding theorem, we mayassume that υ , υ have coordinates in M ( V ), the field of meromorphic functions on V . Itfollows that for some functions φ i : V → U , we can write υ i = υ ( φ i (¯ t )). Now since υ i is asolution to χ ( y ) = ˜ χ (¯ t ) , we have that˜ χ ( φ i (¯ t )) = χ ( υ ( φ i (¯ t ))) = ˜ χ (¯ t ) . From ˜ χ ( φ i (¯ t )) = ˜ χ (¯ t ) it follows that φ i (¯ t ) = g i ¯ t for some g i ∈ G ( C ). DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 24
Finally if we assume there is the set P of polynomials in C [ X, Y ] such that P ( υ , υ ) = 0,then we have that P ( υ ( g ¯ t ) , υ ( g ¯ t )) = 0. Standard arguments using double cosets and thecommensurator Comm G (Γ) of Γ (cf. proof of [14, Lemmas 5.15]) shows that g and g arein the same coset of Comm G (Γ). Hence P is Γ-special. (cid:3) Combining the above with Theorem 5.14 we obtain what can be considered a weak formof the Ax-Lindemann-Weierstrass Theorem with derivatives:
Corollary 5.16.
Assume that ( Y, Y ) is simple. Let υ , . . . , υ n ∈ Y are distinct solutionsthat are not in any Comm G (Γ) -correspondence. Then tr.deg. K K h υ , . . . , υ n i = nk, that is the solutions and their derivatives are algebraically independent over K . We can use Corollary 5.16 and arithmeticity to give a characterization of ω -categoricityof the pregeometry associated with the solution set Y generalizing that given in [14]. Firstwe recall the following deep result of Margulis: Fact 5.17.
Let Γ an irreducible lattice. Then Γ is arithmetic if and only if Γ has infiniteindex in Comm G (Γ) . Corollary 5.18.
Assume that Γ an irreducible lattice and ( Y, Y ) simple. Then Y is non- ω -categorical if and only if Γ is arithmetic. Products of curves
In this section, we pay particular attention to the case dim Y = 1. We aim to showhow all the concepts define in Section 5 and Subsection 3.2 can be explicitly derived in thissituation. We then apply the relevant results to study the fibers of the Schwarzian equationfor Fuchsian groups.6.1. Projective structure on curves.
Consider the group G = PSL ( C ) and its subgroup B of lower triangular matrices so that G/B = CP . A ( G, G/B )-structure on a curve Y isusually called a projective structure (cf.[18]). Let us describe it in the fomalism of Subsection3.2. We consider Y a complex affine algebraic curve defined by a polynomial equation, P ( y, w ) = 0 . Without loss of generality we assume that y is a local coordinate at every point of y ,that is, the differential form dy has no zeroes on Y . The algebraic structure of the jet space J ( Y, CP ) is given by its ring of regular functions and the latter is the D Y -algebra generatedby O Y ⊗ O CP . Let us take t to be the affine coordinate in CP and write the ring of regularfunctions on J ( Y, C ) ⊂ J ( Y, CP ) as C [ Y ][ t, ˙ t, ¨ t, . . . ]The D Y differential structure of this ring is given by the action of ddy = ∂∂y + ˙ t ∂∂t + ¨ t ∂∂ ˙ t + ... t ∂∂ ¨ t + · · · . The open subset J ∗ ( Y, CP ) is the set of jets of submersive maps. It is defined by theinequation ˙ t = 0. Its ring of regular functions is denoted by O J ∗ . To describe the set DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 25 C ⊂ J ∗ ( Y, CP ) of jets of charts of the projective structure, we need to introduce theSchwarzian derivative with respect to the coordinate y , namely S y ( t ) = ... t ˙ t − (cid:16) ¨ t ˙ t (cid:17) . Let R be a rational function on Y and consider in O J ∗ the ddy -ideal generated by S y ( t ) − R. Since S y ( t ) = S y ( t ) if and only if t = at + bct + d for some (cid:18) a bc d (cid:19) ∈ GL ( C ), the local analyticsolutions of this equation are charts of a projective structure. The zero set of this differentialideal is C .Let us give a down to earth description of C and the induced Y . As the equation hasorder 3 and is degree 1 in ... t , C is isomorphic as an algebraic variety to J ∗ ( Y, CP ) andthe D Y -structure induced on its ring of regular maps C [ Y ][ t, ˙ t, t , ¨ t ] is given by the aboveequation ddy = ∂∂y + ˙ t ∂∂t + ¨ t ∂∂ ˙ t + (cid:18)
32 ¨ t ˙ t + 2 R ˙ t (cid:19) ∂∂ ¨ t . The set of uniformizations Y ⊂ J ∗ ( CP , Y ) is isomorphic to J ∗ ( CP , Y ) as an algebraicvariety. We also have that the open subsets J ∗ ( Y, C ) and J ∗ ( C , Y ) are isomorphic asalgebraic varieties. The ring of regular functions on J ∗ ( C , Y ) is C [ t ][ Y ][ y ′ , y ′ , y ′′ , . . . ]and the isomorphism J ∗ ( Y, C ) ≃ J ∗ ( C , Y ) is given by usual formula to express the derivationof a reciprocal function, namely y ′ = t , y ′′ = − ¨ t ˙ t , ...But now notice that as pro-algebraic varieties J ∗ ( Y, CP ) ≃ J ∗ ( CP , Y ) and under thisisomorphism C and Y coincide. Moreover, the differential structures are not the same ason J ∗ ( CP , Y ) the differential structure of the srtuctural ring is given by ddt = ∂∂t + y ′ ∂∂y + y ′′ ∂∂y ′ + y ′′′ ∂∂y ′ + · · · . The subset Y is the zero set of the differential ideal generated by S t ( y ) + 2 Ry ′ = 0. Asalready mentioned Y is isomorphic to J ∗ ( Y, CP ). The ddt -differential structure inducedon its ring of regular maps C [ t ][ Y ][ y ′ , y ′ , y ′′ ] is given by the above equation ddt = ∂∂t + y ′ ∂∂y + y ′′ ∂∂y ′ + y ′′ y ′ − Ry ′ ! ∂∂y ′′ . To describe the connection form observe that the choice of our coordinates on open subset J ∗ ( Y, C ) induces a trivialization, Y × PSL ( C ) → J ∗ ( Y, CP ) , (cid:18) y, w, (cid:20) a bc d (cid:21)(cid:19) ( y, w, t, ˙ t, ¨ t ) DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 26 where t = − b/a , ˙ t = 1 /a , ¨ t = − c/a , or equivalently y ′ = a , y ′′ = 2 ca . We see, by directsubstitution that the linear matrix differential equation in Y × PSL ( C ) ,(6.1) dUdy = A ( y, w ) U where A ( y, w ) = (cid:20) R ( y, w ) 0 (cid:21) and R ( y, w ) is a rational function in Y is equivalent to, S y ( t ) − R ( y, w ) = 0 , S t ( y ) + 2 R ( y, w ) y ′ = 0in the corresponding systems of coordinates on J ∗ ( Y, C ). Let us define the matrix-valuedrational 1-form on J ∗ ( Y, CP ):Ω = U − dU − U − A ( y, w ) U dy
This 1-form Ω is the connection form of the differential equation. It has the followingproperties (some of which were already pointed out in Subsection 2.4):(a) The kernel of Ω is the PSL ( C )-connection F tangent to the graphs of solutions ofthe Schwarzian differential equation.(b) Ω takes values in sl ( C ).(c) If X is the infinitesimal generator of a monoparametric group of right translations { R exp( εB ) : ε ∈ C } for certain B ∈ sl ( C ) then Ω( X ) = B .(d) Ω is adj-equivariant R ∗ g (Ω) = Adj − g ◦ Ω.(e) d Ω + [Ω , Ω] = 0.6.2.
Algebraic relations between solutions of n Schwarzian equations.
Let us con-sider Y , . . . , Y n affine algebraic curves, and for each Y i the bundle J i = J ∗ ( CP , Y i ). Theproduct ˜ J = J × . . . × J n is a (PSL ( C )) n -bundle over the product ˜ Y = Y × . . . × Y n . Letus consider n Schwarzian equations, S t i y i + 2 R i ( y i , w i ) y ′ i = 0 . Each one is seen as a PSL ( C )-invariant connection ∇ i in J i over Y i with connection form Ω i .We consider ˜ ∇ the product ˜ ∇ = ∇ × . . . × ∇ n which is a (PSL ( C )) n invariant connectionon ˜ J over ˜ Y .We are now ready to state the relevant version of Theorem 5.2. Notice that, since we areworking on curves, the factors Y i have no { e } -special subvarieties. Theorem 6.1.
Let ( Y i , ∇ i ) be algebraic curves with simple (PSL ( C ) , CP ) -structures. As-sume that ˆ V : Spf C [[ s , . . . , s k ]] → ˜ J is a non componentwise constant formal parameterized space in a horizontal leaf of ˜ ∇ , andlet V be the Zariski closure of ˆ V . The following are equivalent: (a) dim V < n + rank(ker Ω | V ) . (b) There are two different indices ≤ i < j ≤ n , a curve X ij ⊂ Y i × Y j with both pro-jections dominant, and a PSL ( C ) -bundle isomorphism between f ∗ i ( J i ) and f ∗ j ( J j ) such that f ∗ i Ω i = f ∗ j Ω j . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 27
In particular, under the hypothesis of Theorem 6.1, the coordinates y i and y j are alge-braically dependent on X ij . In other words it follows that if υ i and υ j be solutions of thecorresponding equations then υ i ( t i ( s )) and υ j ( t j ( s )) are algebraically dependent over C .Note that the rank of ker Ω | V is at least the dimension of the smallest analytic subvarietycontaining ˆ V and thus is greater or equal to the rank of the jacobian of our n formal powerseries in k variables. A more precise description of the possible dimension of a non-trivialintersection of V with a leaf is as follows Corollary 6.2.
Let ( Y , ∇ ) , ( Y , ∇ ) be algebraic curves with simple (PSL ( C ) , CP ) -structures. If there exist an horizontal leaf L = L × L ⊂ J × J and an algebraicsubvariety V ⊂ J × J such that dim V ∩ L > and such that V is the Zariski closure ofa positive dimensional irreducible component of V ∩ L , then dim V is , or .Proof. By lemma 3.5, the rank of ker Ω | V can be 1 or 2. From the above theorem, ifdim V <
7, then V is the graph of a gauge correspondence between ∇ and ∇ thusdim V = dim J = dim J = 4. (cid:3) Orthogonality of fibers.
We can now use Theorem 5.14 and Theorem 6.1 to proveseveral key results that were obtained in [14] in the special case of hyperbolic curves ofgenus 0. Our focus remains on the Schwarzian equation( ⋆ ) S t ( y ) + 2 R ( y, w ) y ′ = 0attached to a complex affine algebraic curve Y . So we now work in the context of 1- DCF ,i.e., U is an ordinary differentially closed field. As in Section 4, we assume that U containsan element t such that t ′ = 1 (so U contains the differential field C ( t )).We assume for the remainder of this section that Y is a hyperbolic curve . More precisely,let ˆ Y be a smooth projective completion of the affine curve Y . We assume that we have aFuchsian group Γ ⊂ PSL ( R ) of the first kind, such that if C Γ denotes the set of cusps of Γand H Γ := H ∪ C Γ , then there is a meromorphic mapping j Γ : H Γ → ˆ Y ( C ). The map j Γ iscalled a uniformizer and is an automorphic function for Γ j Γ ( gτ ) = j Γ ( τ ) for all g ∈ Γ and τ ∈ H and so factorizes in a bi-rational isomorphism of Γ \ H Γ into ˆ Y ( C ). We have that j Γ is asolution of the Schwarzian equation ( ⋆ ) for some rational function R .Using [28, Corollary B.1], we have Γ is Zariski dense in Gal( ∇ ), for the correspondingconnection ∇ . So Gal( ∇ ) = G = PSL ( C ). We can hence apply Theorem 5.14, to concludethat the set defined by equation ( ⋆ ) is strongly minimal and satisfies the refined versionof geometric triviality. We also have that equation ( ⋆ ) satisfies the weak for of the Ax-Lindemann-Weierstrass Theorem given in Corollary 5.16. We now aim to recover all theremaining main theorems from the paper [14].For a ∈ U , by a fiber of the Schwarzian equation ( ⋆ ) for a uniformizer j Γ of a Fuchsiangroup Γ of the first kind, we mean an equation of the form χ Γ , ddt ( y ) = a, where χ Γ , ddt ( y ) := S t ( y ) + ( y ′ ) R j Γ ( y, w ) . (6.2) DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 28
The proof of the following result is identical to that of Theorems 6.2 in [14] (see also [20,Proposition 5.2]).
Proposition 6.3.
The set defined by χ Γ , ddt ( y ) = a , with a ∈ U , is strongly minimal andgeometrically trivial. If a , . . . , a n satisfy χ Γ , ddt ( a i ) = a and are dependent, then there exist i, j ≤ n and a Γ -special polynomial P such that P ( a i , a j ) = 0 . For the next results, we will need some more notions from model theory.
Definition 6.4.
Let X and Y be two strongly minimal sets both defined over somedifferential field K ⊂ U .(1) X and Y are nonorthogonal if there is some definable (possibly with additionalparameters) relation R ⊂ X × Y such that the images of the projections of R to X and Y respectively are infinite and these projections are finite-to-one.(2) X and Y are non weakly orthogonal if they are nonorthogonal, that is there is aninfinite finite-to-finite relation R ⊆ X × Y , and the formula defining R can bechosen to be over K alg . Remark . Suppose X and Y (as above) are nonorthogonal and that the relation R ⊂ X × Y witnessing nonorthogonality is defined over some differential field F extending K .Then by definition for any υ ∈ X \ F alg there exist υ ∈ Y \ F alg such that ( υ, υ ) ∈ R .In that case F h υ i alg = F h υ i alg , that is υ, υ and derivatives are algebraically dependentover F .We will need the following important fact. We restrict ourselves to geometrically trivialstrongly minimal sets as this is all we need for the Schwarzian equations. We direct thereader to [33, Corollary 2.5.5] for the more general context. Fact 6.6.
Let X and Y be strongly minimal sets both defined over some differential field K . Assume further that they are both geometrically trivial. If X and Y are nonorthogonal,then they are non weakly orthogonal. Let Γ and Γ be two Fuchsian groups. We say that Γ is commensurable with Γ in wide sense if Γ is commensurable to some conjugate of Γ . We will now answer a questionthat was left open in [14], where only the case a = b = 0 (and genera 0) was established. Theorem 6.7.
Let Γ and Γ be two Fuchsian groups of the first kind. Assume furtherthat Γ is not commensurable with Γ in the wide sense. For a, b ∈ U , the strongly minimalsets defined by equations χ Γ , ddt ( y ) = a and by χ Γ , ddt ( y ) = b are orthogonal. We will need the following lemma, a weaker form of which is Theorem 6.3 of [14]. Weuse the notation χ − , ddt ( a ) for the set defined by χ Γ , ddt ( y ) = a . Lemma 6.8.
Let Γ and Γ be two Fuchsian groups of the first kind. For a = b , the stronglyminimal sets χ − , ddt ( a ) and χ − , ddt ( b ) are orthogonal.Proof. Throughout, we respectively use M ( U ) and D ( p, r ) for the field of meromorphicfunctions on a domain U ⊂ C , and the open complex disk centered at p ∈ C with ra-dius r. As both χ − , ddt ( a ) and χ − , ddt ( b ) are strongly minimal and geometrically trivial, if DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 29 χ − , ddt ( a ) χ − , ddt ( b ), then there is a finite-to-finite correspondence between the sets, de-fined over Q h a, b i . Using Seidenberg’s embedding theorem, we regard a, b as meromorphicfunctions on a domain U ⊂ C . In what follows ˜ a denotes a meromorphic function such that S ddt (˜ a ) = a . The function ˜ b is defined similarly.Using the holomorphic inverse function theorem, we claim that without loss of gener-ality, it is enough to prove the result for the case a = 0. Indeed, since j Γ (˜ a ( t )) is inter-algebraic with j Γ ( g ˜ b ( t )) for some g ∈ GL ( C ), we have that j Γ ( t ) is interalgebraic with j Γ ( g ˜ b (˜ a − ( t )) (since ˜ b is defined up to composition with linear fractional transformations,we can assume that there is a common regular point for ˜ a and ˜ b and work locally aroundthis point). Letting ˜ c = ˜ b ◦ ˜ a − and c = S ddt (˜ c ), we see that χ − , ddt (0) χ − , ddt ( c ) and bygeometric triviality this occurs over Q h c i .So we assume that a = 0. Let p be a regular point for ˜ b ( t ) and let D = D ( p, ǫ ) bea disc of regular points of ˜ b ( t ). Also let γ be a linear fractional transformation sending D = D ( p, ǫ ) to H .Since χ − , ddt (0) χ − , ddt ( b ), we have that for some g ∈ GL ( C ), the solution j Γ ( g ˜ b ( t )) isalgebraic over Q h b, j Γ ( γt ) i ⊂ M ( D )( j Γ ◦ γ, j ′ Γ ◦ γ, j ′′ Γ ◦ γ ) ⊂ M ( D ). But notice that forany domain U such that D ⊆ U ⊆ D , if j Γ ( g ˜ b ( t )) is algebraic over M ( U ), then j Γ ( γt )will also be algebraic over M ( U ). This follows from the fact that M ( D ) ⊆ M ( U ), and j Γ ( g ˜ b ( t )) is interalgebraic with j Γ ( γt ) over Q h b i ⊂ M ( D ). But j Γ ( t ) cannot be extendedalgebraically on a neighborhood of H , hence U = D .The disc D is thus the maximal among domains U such that j Γ ( g ˜ b ( t )) is algebraicover M ( U ). But such a domain satisfies g ˜ b ( D ) = H , that is the image of D by theregular holomorphic map ˜ b is the disc g − H . A corollary of Schwarz’s lemma gives thatbiholomorphisms from a disc to a disc are restrictions of homographies. Hence ˜ b is anhomography h ∈ PSL ( C ) and so b = 0. (cid:3) Proof of Theorem 6.3.
By Lemma 6.8, we may assume that a = b and so we need to classifynon-orthogonalities between the strongly minimal sets defined by χ Γ , ddt ( y ) = a and by χ Γ , ddt ( y ) = a . As easily verified (cf. [14, Lemma 6.1]), if we let K = Q h a i and let ∂ = 1˜ a ′ ddt , then we have that K is a ∂ -differential field and for each i = 1 ,
2, the sets χ Γ i , ddt ( y ) = a and χ Γ i ,∂ ( y ) = 0 coincide. Hence χ Γ , ddt ( y ) = a is non-orthogonal to by χ Γ , ddt ( y ) = a if andonly if χ Γ ,∂ ( y ) = 0 is non-orthogonal to χ Γ ,∂ ( y ) = 0. We can apply [14, Theorem 6.5],which state that if Γ is not commensurable with Γ in the wide sense, then χ Γ ,∂ ( y ) = 0and χ Γ ,∂ ( y ) = 0 are orthogonal. (cid:3) Finally, without giving further details, let us mention that one can also establish the Ax-Lindemann-Weierstrass Theorem with derivatives for j Γ using the same arguments from[14]. However, this also follows from the Ax-Schanuel theorem below. One can also followthe strategy given in [14] to obtain a special case of the Andr´e-Pink conjecture for j Γ . DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 30 Proof of the Ax-Schanuel Theorems for products of curves.
In this section, we prove two instances of the Ax-Schanuel Theorem with derivatives.First we tackle the case of products of hyperbolic curves, a direct generalization of the work[31] and [27] in the case of curves. We then prove the Ax-Schanuel Theorem with derivativesfor non-hyperbolic curves given from “generic triangle groups”.Let us state the general problem for a Schwarzian equation. Let υ denote a solution of aSchwarzian equation( ⋆ ) S t ( y ) + 2 R ( y, w ) y ′ = 0attached to a complex affine algebraic curve Y and such that Gal( ∇ ) = PSL ( C ). Noticethat we take υ : U → Y to be a uniformization function as defined in Subsection 3.2. Letˆ t , . . . , ˆ t n be formal parameterizations of neighborhoods of points p , . . . p n in U . We write δ i for the derivations induced by differentiation with respect to ˆ t i . The Ax-Schanuel Theoremwith derivatives for υ is an answer to the following problem. Problem.
Fully characterize the conditions on ˆ t , . . . , ˆ t n for which tr . deg . C C (ˆ t , υ (ˆ t ) , υ ′ (ˆ t ) , υ ′′ (ˆ t ) , . . . , ˆ t n , υ (ˆ t n ) , υ ′ (ˆ t n ) , υ ′′ (ˆ t n )) < n + rank( δ i ˆ t j ) . Using Theorem 6.1 (which holds since Gal( ∇ ) = PSL ( C )) with Y = Y i and υ = υ i for i =1 , . . . , n , to prove the Ax-Schanuel Theorem with derivatives we only need to characterizethe conditions on pairs ˆ t , ˆ t . Moreover, from Corollary 6.2, if ˆ t , ˆ t are nonconstant andtr . deg . C (cid:0) C (ˆ t , ˆ t , υ (ˆ t ) , υ ′ (ˆ t ) , υ ′′ (ˆ t ) , υ (ˆ t ) , υ ′ (ˆ t ) , υ ′′ (ˆ t ) (cid:1) = ℓ = 7 or 8 , then it must be that ℓ = 4 and there is a polynomial P ∈ C [ x, y ] such that P ( υ (ˆ t ) , υ (ˆ t )) =0 . The case of hyperbolic curves.
We prove the Ax-Schanuel Theorem with deriva-tives for all hyperbolic curves. Let Γ ⊂ PSL ( R ) be a Fuchsian group of the first kind andlet j Γ be a uniformizing function for Γ. Theorem 7.1.
Let ˆ t , . . . , ˆ t n be formal parameterizations of neighborhoods of points p , . . . p n in H . Assume that ˆ t , . . . , ˆ t n are geodesically independent, namely ˆ t i is nonconstant for i = 1 , . . . , n and there are no relations of the form ˆ t i = γ ˆ t j for i = j , i, j ∈ { , . . . , n } and γ is an element of Comm (Γ) , the commensurator of Γ . Then tr . deg . C C (ˆ t , j Γ (ˆ t ) , j ′ Γ (ˆ t ) , j ′′ Γ (ˆ t ) , . . . , ˆ t n , j Γ (ˆ t n ) , j ′ Γ (ˆ t n ) , j ′′ Γ (ˆ t n )) ≥ n + rank( δ i ˆ t j ) . Theorem 7.1 follows from the following lemma.
Lemma 7.2.
Assume that tr . deg . C (cid:0) C (ˆ t , ˆ t , j Γ (ˆ t ) , j ′ Γ (ˆ t ) , j ′′ Γ (ˆ t ) , j Γ (ˆ t ) , j ′ Γ (ˆ t ) , j ′′ Γ (ˆ t ) (cid:1) = 4 , then the polynomial P such that P ( j (ˆ t ) , j (ˆ t )) = 0 is a Γ -special polynomial (and so ˆ t and ˆ t are geodesically dependent).Proof. To simplify notation we write j = j Γ and t i instead of ˆ t i . Consider the derivation δ and let us write χ Γ ,δ ( x ) = a , where a ∈ C h t i for the Schwarzian equation of j ( t ) (so χ Γ ,δ ( x ) = 0 is the equation of j ( t )). DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 31
Claim : j ( t ) , j ( t ) / ∈ C h t i alg . Proof of Claim.
Assume for some i , j ( t i ) ∈ C h t i alg . Then for both i = 1 , j ( t i ) ∈ C h t i alg ; this follows because j ( t ) and j ( t ) are interalgebraic over C . Now, notethat we have that tr . deg . C ( C h t , j ( t ) i ) = 4. Using the assumption ℓ = 4 we get that t ∈ C h t , j ( t ) i alg (of course so is j ( t )). Hence, it must be the case that C h t , t i alg = C h t , j ( t ) i alg = C h t , j ( t ) i alg . The last equality is obtained from the interalgebraicity of j ( t ) and j ( t ) over C and from the fact that j ( t ) C ( t ) alg . Hence we have that j ( t )and t are interalgebraic over the field C ( t ) and that tr . deg . C ( C h t , t i ) = 4.But notice that t , j ( t ), j ′ ( t ), j ′′ ( t ) are algebraically independent over C . So, the as-sumption j ( t ) ∈ C h t i alg and tr.deg. C ( C h t , t i ) = 4 gives that tr . deg . C ( C h t i ) = 4 andthat t ∈ C h t i alg . But as observed above, we have that t is interalgebraic with j ( t ) over C ( t ). It hence follows that t is interalgebraic with j ( t ) over C , since t ∈ C h t i alg . Thiscontradicts the fact that tr . deg . C ( C h j ( t ) i ) = 3. (cid:3) So from the claim and strong minimality of the two equations, we have thattr . deg . C h t i ( C h j ( t i ) i ) = 3for both i = 1 and i = 2. That is j ( t ) and j ( t ) are generic (over C h t i ) solutions of theirrespective equations. Since P ( j ( t ) , j ( t )) = 0, we get that generic solutions of χ Γ ,δ ( x ) = a and χ Γ ,δ ( x ) = 0 are interalgebraic over C h t i . So the two equations are non-orthogonal.Using Lemma 6.8 with Γ = Γ = Γ we get that a = 0 and from Corollary 5.16 we havethat P is a Comm (Γ)-correspondence. (cid:3)
We hence also answer positively a question of Aslanyan [2, Section 3.4] about the existenceof differential equations satisfying the Ax-Schanuel Theorem with derivatives and such thatthe polynomial ( X − Y ) is the only Γ-special polynomial. Indeed, this is true of any Γsatisfying Γ = Comm(Γ). Many such examples exist (cf. [12, Fact 4.9]).7.2. The case of generic triangle groups.
We will now exploit the fact that the proofof Lemma 7.2 only depends on the following:(1) Theorem 6.1 and Corollary 6.2 hold for χ Γ , ddt ( y ) = 0, i.e., Gal( ∇ ) = PSL ( C );(2) There is a full characterization of the structure of the set defined by any fiber of theSchwarzian equation χ Γ , ddt ( y ) = a ; and(3) If a = b , then the set defined by χ Γ , ddt ( y ) = a and χ Γ , ddt ( y ) = b are orthogonal.We can hence apply this technique to any context where the above three properties havebeen established. As it turns out, in [12], property 1 and 2 was established for generictriangle groups. We now recall this setting and show that property 3 also holds and henceobtain the Ax-Schanuel with derivatives.Let △ ⊂ H be an open circular triangle with vertices v , v , v and with respective internalangles πα , πβ and πγ . Using the Riemann mapping theorem, we have a unique biholomorphicmapping J : △ → H sending the vertices v , v , v to ∞ , , J ( t ) to a homeomorphism from the closure of △ onto H = H ∪ P ( R ). The function DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 32 J ( t ) is called a Schwarz triangle function and is a uniformization function in the sense ofSubsection 3.2. It satisfies the Schwarzian equation ( ⋆ ) R ( y, w ) = R α,β,γ ( y ) = 12 (cid:18) − β − y + 1 − γ − ( y − + β − + γ − − α − − y ( y − (cid:19) . We call this equation a Schwarzian triangle equation and write it as χ α,β,γ, ddt ( y ) = 0 . By ageneric such equation we mean the case when α, β, γ ∈ R > are algebraically independentover Q . Remark . In [12], the parameters α, β, γ where allowed to be arbitrary complex numbers.However, to apply Theorem 6.1 (curve case), we require that α, β, γ ∈ R > .Let us now state what is known about generic Schwarzian triangle equations. The proofcan be found in [12, Section 4]. We denote χ α,β,γ, ddt ( y ) = a for the fiber equations. Fact 7.4.
Let α, β, γ ∈ R be algebraically independent over Q and let a ∈ U be arbitrary.Then (1) The Galois group
Gal ( ∇ ) for the corresponding connection ∇ is PSL ( C ) . (2) The set defined by χ α,β,γ, ddt ( y ) = a is strongly minimal, geometrically trivial andstrictly disintegrated. Namely, if K is any differential field extension of Q ( α, β, γ ) h a i and y , . . . , y n are distinct solutions that are not algebraic over K , then tr . deg . K K( y , y ′ , y ′′ , . . . , y n , y ′ n , y ′′ n ) = 3 n . We only need to prove orthogonalities of the distinct fiber equations.
Proposition 7.5.
Let α, β, γ ∈ R be algebraically independent over Q . Let a, b ∈ U be suchthat a = b . Then the strongly minimal sets defined by χ α,β,γ, ddt ( y ) = a and χ α,β,γ, ddt ( y ) = b are orthogonal.Proof. Assume that α, β, γ ∈ R are algebraically independent over Q . We denote by C thefield of constants generated by α, β, γ over Q , that is C = Q ( α, β, γ ). For u, v, w, d ∈ U , wedenote by X ( u, v, w, d ) the set defined by a fiber equation χ u,v,w, ddt ( y ) = d and u ′ = v ′ = w ′ = 0. Notice that is a uniformly defined family.Arguing by contradiction, assume that X ( α, β, γ, a ) and X ( α, β, γ, b ) are non-orthogonal.Since the sets are geometrically trivial, we have that they are non-weakly orthogonal. Sowe have a definable relation R ⊂ X ( α, β, γ, a ) × X ( α, β, γ, b ) defined over C h a, b i alg . Let R a and R b be the respective projections of R to X ( α, β, γ, a ) and X ( α, β, γ, b ). By defi-nition, R a and R b are finite sets defined over C h a, b i alg . We define Y ( α, β, γ, a ) to be thecomplement X ( α, β, γ, a ) \ R a . The set Y ( α, β, γ, b ) is defined similarly.Then it is not hard to see that we have an L -formula θ ( u , u , u ) over Q such that θ ( α, β, γ ) is the sentence asserting that ∃ a, b a = b ( ∀ x, y ( x ∈ Y ( α, β, γ, a ) ∧ y ∈ Y ( α, β, γ, b )) = ⇒ ( x, y ) ∈ R ) . It follows by construction that if θ ( α , β , γ ) holds in U , that is U | = θ ( α , β , γ ), then α , β , γ ∈ C and there is a , b ∈ U such that a = b and the definable sets X ( α , β , γ , a ) DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 33 and X ( α , β , γ , b ) are non-orthogonal. Using the Fact [12, Fact 2.2] with V = A and F = Q , since α, β, γ are algebraically independent over Q , we can find k, l, m ∈ N suchthat 2 ≤ k ≤ l ≤ m and U | = θ ( k, l, m ). So there is a , b ∈ U such that a = b andthe definable sets X ( k, l, m, a ) and X ( k, l, m, b ) are non-orthogonal. But for any d ∈ U ,the set X ( k, l, m, d ) is the set defined by the fiber of the Schwarzian equation ( ⋆ ) for theuniformizer of the Fuchsian (triangle) group Γ ( k,l,m ) with signature (0; k, l, m ). Hence thiscontradicts Theorem 6.7 with Γ = Γ = Γ ( k,l,m ) . (cid:3) We now assume that α, β, γ ∈ R > are algebraically independent over Q and let J : △ → H be the corresponding Schwarz triangle function as describe above. Theorem 7.6.
Let ˆ t , . . . , ˆ t n be distinct formal parameterizations of neighborhoods of points p , . . . p n in △ . Then tr . deg . C C (ˆ t , J (ˆ t ) , J ′ (ˆ t ) , J ′′ (ˆ t ) , . . . , ˆ t n , J (ˆ t n ) , J ′ (ˆ t n ) , J ′′ (ˆ t n )) ≥ n + rank( δ i ˆ t j ) . Theorem 7.6 follows from the following lemma.
Lemma 7.7.
Assume that tr . deg . C (cid:0) C (ˆ t , ˆ t , J (ˆ t ) , J ′ (ˆ t ) , J ′′ (ˆ t ) , J (ˆ t ) , J ′ (ˆ t ) , J ′′ (ˆ t ) (cid:1) = 4 , then J (ˆ t ) = J (ˆ t ) and hence ˆ t = ˆ t .Proof. As in the proof of Lemma 7.2, we write t i instead of ˆ t i and consider the derivation δ and write χ α,β,γ,δ ( x ) = a , where a ∈ C h t i for the Schwarzian equation of J ( t ) (so χ α,β,γ,δ ( x ) = 0 is the equation of J ( t )). It then follows (using the exact same argument)that J ( t ) , J ( t ) / ∈ C h t i alg . Hence using strong minimality of the two equations, we havethat tr.deg. C h t i ( C h J ( t i ) i ) = 3 for both i = 1 and i = 2. So J ( t ) and J ( t ) are generic(over C h t i ) solutions of their respective equations.Since tr . deg . C ( C ( t , t , J ( t ) , J ′ ( t ) , J ′′ ( t ) , J ( t ) , J ′ ( t ) , J ′′ ( t )) = 4, we have some poly-nomial P ∈ C [ X, Y ] such that P ( J ( t ) , J ( t )) = 0. So generic solutions of χ α,β,γ,δ ( x ) = a and χ α,β,γ,δ ( x ) = 0 are interalgebraic over C h t i . Hence the two equations are non-orthogonal. Using Proposition 7.5 we get that a = 0 and from Fact 7.4(2) we have that P is the polynomial ( X − Y ). So the result follows. (cid:3) References [1] V. Aslanyan,
Adequate predimension inequalities in differential fields , arXiv preprint arXiv:1803.04753(2018).[2] V. Aslanyan,
Ax-Schanuel and strong minimality for the j -function , Ann. Pure Appl. Logic 172 (2021),no. 1, 102871, 24 pp.[3] V. Aslanyan, Weak modular Zilber-Pink with derivatives , arXiv preprint arXiv:1803.05895, 2018.[4] V. Aslanyan, S. Eterovi´c, J. Kirby.
A closure operator respecting the modular j -function , arXiv preprintarXiv:2010.00102 (2020).[5] V. Aslanyan, S. Eterovi´c, J. Kirby. Differential Existential Closedness for the j -function , Proc. Amer.Math. Soc. (2021) https://doi.org/10.1090/proc/15333.[6] V. Aslanyan, J. Kirby. Blurrings of the j -function , arXiv preprint arXiv:2005.10167 (2020).[7] J. Ax. On Schanuel’s conjectures , Annals of Mathematics, 93:252-268, 1971.[8] U. Bader and D. Fisher and N. Miller and M. Stover
Arithmeticity, superrigidity and totally geodesicsubmanifolds of complex hyperbolic manifolds arXiv preprint arXiv:2006.03008. 2020.
DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 34 [9] B. Bakker, J. Tsimermann,
Lectures on the Ax-Schanuel Conjecture
Arithmetic Geometry of Logarith-mic Pairs and Hyperbolicity of Moduli Spaces, CRM Short Courses, Springer[10] G. Baldi, Emmanuel Ullmo,
Special subvarieties of non-arithmetic ball quotients and Hodge Theory ,arXiv preprint arXiv:2005.03524 (2020).[11] F. Barroero and G. Dill, On the Zilber-Pink conjecture for complex abelian varieties, arXiv preprintarXiv:1909.01271, 2019.[12] D. Bl´azquez Sanz, G. Casale, J. Freitag and J. Nagloo,
Some functional transcendence results aroundthe Schwarzian differential equation to appear in the Annales de la Facult´e des Sciences de Toulouse -Volume in honor of Prof. H. Umemura.[13] P. Bonnet,
Minimal invariant varieties and first integrals for algebraic foliations , Bull. Braz. Math. Soc.(N.S.) 37 (2006), no. 1, p. 1-17.[14] G. Casale, J. Freitag and J. Nagloo,
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuch-sian groups , Ann. of Math. (2) 192 (2020), no. 3, 721-765.[15] C. Daw and J. Ren,
Applications of the hyperbolic Ax–Schanuel conjecture , Compositio Mathematica154 (2018), no. 9 1843–1888.[16] T. Driscoll and L. Trefethen,
Schwarz-christoffel mapping.
Vol. 8. Cambridge University Press, 2002.[17] P. Deligne and G.D. Mostow,
Monodromy of hypergeometric functions and non-lattice integral mon-odromy
Publications Math´ematiques de l’IH´ES, Tome 63 (1986): 5-89.[18] D. Dumas,
Complex projective structures.
Handbook of Teichm¨uller theory. Vol. II, 455-508, IRMALect. Math. Theor. Phys., 13, Eur. Math. Soc., Z¨urich, 2009.[19] M. Epstein, M. Elzanowski,
Material Inhomogeneities and their Evolution, A Geometric Approach
Interaction of Mechanics and Mathematics, Springer-Verlag Berlin Heidelberg 2007[20] J. Freitag and T. Scanlon.
Strong minimality and the j -function . Journal of the European MathematicalSociety 20.1 (2017): 119-136.[21] S. Helgason, Differential geometry, Lie groups, and symmetric spaces . Graduate Studies in Mathematics,Vol. 34, Amer. Math. Soc., Providence, RI, 2001.[22] Kirby, Jonathan, and Boris Zilber,
Exponentially closed fields and the conjecture on intersections withtori.
Annals of Pure and Applied Logic 165.11 (2014): 1680-1706.[23] B. Klingler, E. Ullmo, and A. Yafaev, The hyperbolic Ax-Lindemann-Weierstrass conjecture, Publ.Math. Inst. Hautes ´Etudes Sci. 123 no. 1 (2016) 333-360.[24] T. McGrail,
The model theory of differential fields with finitely many commuting derivations,
J. SymbolicLogic 65 (2000), no. 2, 885-913.[25] N. Mok,
Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finitevolume.
In Perspectives in analysis, geometry, and topology, volume 296 of Progr. Math., pages 331-354.Springer, New York 2012.[26] N. Mok,
Zariski closures of images of algebraic subsets under the uniformization map on finite-volumequotients of the complex unit ball.
Compositio Mathematica, volume 155, no. 11, pages 2129–2149.Cambridge University Press, 2019.[27] N. Mok, J. Pila, J. Tsimerman, Ax-Schanuel for Shimura varieties, Annals of Mathematics 189.3 (2019):945-978.[28] J. Morales Ruiz,
Differential Galois theory and non-integrability of Hamiltonian systems , Progress inMathematics, 179. Birkh¨auser Verlag, Basel, 1999.[29] J. Pila, Rational Points of Definable Sets and Results of Andr´e-Oort-Manin-Mumford type, InternationalMathematics Research Notices 2009.13 (2009): 2476-2507.[30] J. Pila, o-minimality and the Andr´e-Oort conjecture for C n , Annals of mathematics (2011): 1779-1840.[31] J. Pila and J. Tsimerman, Ax-Schanuel for the j -function, Duke Math. J. 165, no. 13 (2016) 2587-2605.[32] J. Pila and J. Tsimerman, The Andr´e-Oort conjecture for the moduli space of abelian surfaces . Compo-sitio Mathematica 149.2 (2013): 204-216.[33] A. Pillay,
Geometric stability theory , No. 32. Oxford University Press, 1996.[34] T. Scanlon,
Algebraic differential equations from covering maps
Adv. Math. 330 (2018), 1071-1100.[35] R. W. Sharpe,
Differential geometry: Cartan’s generalization of Klein’s Erlangen program , Springer(2000)
DIFFERENTIAL APPROACH TO THE AX-SCHANUEL, I 35 [36] J. Tsimerman,
The Andr´e-Oort conjecture for A g . Ann. of Math.(2) 187.2 (2018): 379-390.[37] E. Ullmo and A. Yafaev, A characterization of special subvarieties, Mathematika 57.2 (2011): 263-273.[38] M. Yoshida, Fuchsian Differential Equations With Special Emphasis on the Gauss-Schwarz Theory
Aspects of Mathematics
David Bl´azquez-Sanz, Universidad Nacional de Colombia - Sede Medell´ın, Facultad de Cien-cias, Escuela de Matem´aticas, Colombia
Email address : [email protected] Guy Casale, Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
Email address : [email protected] James Freitag, University of Illinois Chicago, Department of Mathematics, Statistics, andComputer Science, 851 S. Morgan Street, Chicago, IL, USA, 60607-7045.
Email address : [email protected] Joel Nagloo, CUNY Bronx Community College, Department of Mathematics and ComputerScience, Bronx, NY 10453, and CUNY Graduate Center, Ph.D. programs in Mathematics, 365Fifth Avenue, New York, NY 10016, USA
Email address ::