aa r X i v : . [ m a t h . C V ] M a y BLURRINGS OF THE j -FUNCTION VAHAGN ASLANYAN AND JONATHAN KIRBY
Abstract.
Inspired by the idea of blurring the exponential function, we defineblurred variants of the j -function and its derivatives, where blurring is givenby the action of a subgroup of GL ( C ) . For a dense subgroup (in the complextopology) we prove an Existential Closedness theorem which states that all sys-tems of equations in terms of the corresponding blurred j with derivatives havecomplex solutions, except where there is a functional transcendence reason whythey should not. For the j -function without derivatives we prove a stronger the-orem, namely, Existential Closedness for j blurred by the action of a subgroupwhich is dense in GL +2 ( R ) , but not necessarily in GL ( C ) .We also show that for a suitably chosen countable algebraically closed subfield C ⊆ C , the complex field augmented with a predicate for the blurring of the j -function by GL ( C ) is model theoretically tame, in particular, ω -stable andquasiminimal. Introduction
Let H be the complex upper half-plane and let j : H → C be the modular j -function. It is invariant under the action of SL ( Z ) on H and behaves nicely underthe action of GL +2 ( Q ) , namely, it satisfies certain algebraic “functional equations”given by the modular polynomials (see Section 2.1). This gives rise to the ModularSchanuel conjecture (see [Pil15, Conjecture 8.3]), henceforth referred to as MSC,which is a transcendence statement about the values of the j -function, and is ananalogue of Schanuel’s conjecture for the exponential function [Lan66, p. 30]. Conjecture 1.1 (Modular Schanuel Conjecture) . Let z , . . . , z n ∈ H be non-quadratic numbers with distinct GL +2 ( Q ) -orbits. Then td Q Q ( z , . . . , z n , j ( z ) , . . . , j ( z n )) ≥ n. This conjecture can be understood as a statement about certain “overdeter-mined” systems of polynomial equations of n variables not having solutions ofthe form ( z , . . . , z n , j ( z ) , . . . , j ( z n )) with z k ’s having pairwise distinct GL +2 ( Q ) -orbits. There is also a “dual” conjecture, known as Existential Closedness , orconcisely EC, stating that if such a system is not overdetermined, that is, havinga solution does not contradict MSC, then there is a solution in C . The notion of Date : May 21, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Ax-Schanuel, Existential Closedness, j -function.Supported by EPSRC grant EP/S017313/1. an overdetermined system of equations in n variables is captured by some proper-ties of an algebraic variety V ⊆ C n known as j - broadness and j - freeness . Roughly, j -broadness means that certain projections of V are not too small, in particular, dim V ≥ n , while j -freeness means that no modular polynomial vanishes on theappropriate coordinates of V . We now formulate the EC conjecture postponingthe precise definitions of j -broadness and j -freeness to Section 2.2.Throughout the paper algebraic varieties will be identified with the sets of their C -points. By a subvariety of H n × C n we mean a non-empty intersection of analgebraic variety in C n with H n × C n . By abuse of notation we will let j denoteall Cartesian powers of itself. Similarly we let Γ j := { (¯ z, j (¯ z )) : ¯ z ∈ H } ⊆ C n bethe graph of j in H n × C n for any n . Conjecture 1.2 (Existential Closedness for j ) . Let V ⊆ H n × C n be an irreducible j -broad and j -free variety defined over C . Then V ∩ Γ j = ∅ . While MSC is considered out of reach, EC seems to be more tractable. Inparticular, EC was proven under the additional assumption that the projection of V on the first n coordinates has dimension n (which in fact implies j -broadness) byEterović and Herrero in [EH20] (and some other related results were also obtainedthere), and a functional (differential) analogue of EC was established in [AEK20]by Aslanyan, Eterović, Kirby. This paper is devoted to proving some weak versionsof EC.There are many analogies between the j -function and the complex exponenti-ation, especially from a transcendence point of view. In particular, MSC is themodular analogue of the famous Schanuel conjecture for exp , and the EC conjec-ture for j is an analogue of a similar conjecture for exp , known as ExponentialClosedness , posed by Zilber in the early 2000’s [Zil04]. Moreover, the aforemen-tioned theorems proven in [EH20] are analogous to some results on the exponentialfunction established in [BM17, DFT18], though the methods are quite different.Similarly, the work in this paper is related to previous work on the blurred complexexponentiation [Kir19], but we also use some new tools here such as combiningmethods from complex and real analytic geometry and o-minimality.
Definition 1.3.
Given a subgroup G ⊆ GL ( C ) let B Gj ⊆ C be the relation { ( z, j ( gz )) : g ∈ G, gz ∈ H } . By abuse of notation we also let B Gj denote theset { ( z , . . . , z n , j ( g z ) , . . . , j ( g n z n )) : g k ∈ G, g k z k ∈ H for all k } for every n .We think of B Gj as the graph of j blurred by the action of G . The following isone of our main results. LURRINGS OF THE j -FUNCTION 3 Theorem 1.4. If V ⊆ C n is a j -broad and j -free variety and G is dense in GL ( C ) in the complex topology then V ∩ B Gj is dense in V , and hence it is non-empty. In fact we prove a more general theorem incorporating the (first two) derivativesof the j -function. First, we state the EC conjecture for j and its derivatives. Let J : H → C be given by J : z ( j ( z ) , j ′ ( z ) , j ′′ ( z )) . We extend J to H n by defining J : ¯ z ( j (¯ z ) , j ′ (¯ z ) , j ′′ (¯ z )) where j ( k ) (¯ z ) = ( j ( k ) ( z ) , . . . , j ( k ) ( z n )) for k = 0 , , . Let Γ J ⊆ H n × C n be thegraph of J for any n . Note that we consider only the first two derivatives of j ,for the higher derivatives are algebraic over those (see [Mah69] and also Section5.1), and from a transcendence point of view adding higher derivatives would notchange anything. Conjecture 1.5 (Existential Closedness for J ) . Let V ⊆ H n × C n be an irre-ducible J -broad and J -free variety defined over C . Then V ∩ Γ J = ∅ . Here J - broadness and J - freeness are analogues of j -broadness and j -freenessrespectively (see Section 2.2 for definitions). Note that the aforementioned differ-ential EC statement for the j -function does in fact incorporate the derivatives of j and so it is a differential variant of Conjecture 1.5 (see [AEK20] for details). Definition 1.6.
For a subgroup G ⊆ GL ( C ) define a relation B GJ := (cid:26)(cid:18) z, j ( gz ) , ddz j ( gz ) , d dz j ( gz ) (cid:19) : g ∈ G, gz ∈ H (cid:27) ⊆ C n . By abuse of notation for each n we denote the set { (¯ z, ¯ w, ¯ w , ¯ w ) : ( z k , w k , w ,k , w ,k ) ∈ B GJ for all k } by B GJ .We prove the following Existential Closedness statement for the blurred J -function. Theorem 1.7.
Let V ⊆ C n be an irreducible J -broad and J -free variety definedover C , and let G ⊆ GL ( C ) be a subgroup which is dense in the complex topology.Then V ∩ B GJ is dense in V in the complex topology. In particular, if C ⊆ C is asubfield with C * R , then V ∩ B GL ( C ) J = ∅ . The proof of this result is based on the Ax-Schanuel theorem for the j -function[PT16] and the Remmert open mapping theorem from complex analytic geometry.It is an adaptation of Kirby’s proof of EC for the blurred exponentiation [Kir19, VAHAGN ASLANYAN AND JONATHAN KIRBY
Proposition 6.2], however dealing with j and its derivatives makes the proof slightlymore involved.Furthermore, we establish a stronger result for j without derivatives. When G = GL +2 ( Q ) we call B Gj the approximate j -function and denote it by A j . Theorem 1.8.
Let V ⊆ H n × C n be an irreducible j -broad and j -free variety de-fined over C and let G ⊆ GL +2 ( R ) be a dense subgroup (in the Euclidean topology).Then V ∩ B Gj is dense in V in the complex topology. In particular, V ∩ A j is densein V . Note that the direct analogue of the approximate exponentiation from [Kir19]would in fact be the relation B GL ( Q ( i )) j , and A j is a finer blurring. EC for B GL ( Q ( i )) j is given by Theorem 1.4, while Theorem 1.8 gives a stronger result. To prove itwe invoke some tools from o-minimality along with basic complex geometry andthe Ax-Schanuel theorem. The analogue of Theorem 1.8 for blurred exp is notknown and our methods would fail there. This is a significant difference betweenthis paper and [Kir19].It is also worth noting that in fact we prove a stronger form of Theorem 1.8.Namely, we show that if G is dense in a certain real Lie subgroup of GL ( R ) ofdimension , then the conclusion of the theorem holds. Moreover, our proof willshow that the theorem also holds for G = SL ( Q ) and even for the subgroupthereof consisting of the upper triangular matrices.Finally, at the end of the paper we show that for a suitably chosen countable al-gebraically closed field C ⊆ C the structures (cid:16) C ; + , · , B GL ( C ) j (cid:17) and (cid:16) C ; + , · , B GL ( C ) J (cid:17) are model theoretically tame. More precisely, we prove that these structures are el-ementarily equivalent to certain reducts of differentially closed fields, which gives acomplete axiomatisation of their theories and shows that they are ω -stable of Mor-ley rank ω and near model complete (i.e. every formula is equivalent to a Booleancombination of existential formulas). We also show that they are quasiminimal ,that is, every definable set (in one variable) is either countable or co-countable.2. Preliminaries
Modular polynomials and special varieties.
The group GL ( C ) acts onthe Riemann sphere by linear fractional transformations and its subgroup GL +2 ( R ) – the group of × real matrices with positive determinant – acts on H . Thisinduces actions of subgroups of GL +2 ( R ) on H . The j -function satisfies certainalgebraic “functional equations” under the action of GL +2 ( Q ) . More precisely,there is a countable collection of irreducible polynomials Φ N ∈ Z [ X, Y ] ( N ≥ ,called modular polynomials , such that for any z , z ∈ H Φ N ( j ( z ) , j ( z )) = 0 for some N iff z = gz for some g ∈ GL +2 ( Q ) . LURRINGS OF THE j -FUNCTION 5 In particular, if τ ∈ H is a quadratic number then j ( τ ) is algebraic. Thesenumbers are known as special values of j or as singular moduli (they are the j -invariants of elliptic curves with complex multiplication). We refer the reader to[Lan73] for details. Definition 2.1. A special subvariety of C n (with coordinates ¯ w ) is an irreduciblecomponent of a variety defined by modular equations, i.e. equations of the form Φ N ( w k , w l ) = 0 for some ≤ k, l ≤ n where Φ N is a modular polynomial.2.2. Broad and free varieties.Notation.
We will use the following notation. • For a positive integer n the tuple (1 , . . . , n ) is denoted by ( n ) , and ¯ k ⊆ ( n ) means that ¯ k = ( k , . . . , k l ) for some ≤ k < . . . < k l ≤ n . • The coordinates of C n will be denoted by ( z , . . . , z n , w , . . . , w n ) . Theprojection on the last n coordinates will be denoted by pr w . Depending onthe context, we will use ¯ z or ¯ w for the coordinates of C n . • For a tuple ¯ k = ( k , . . . , k l ) ⊆ ( n ) define the projection map π ¯ k : C n → C l by π ¯ k : ( z , . . . , z n ) ( z k , . . . , z k l ) . Further, define pr ¯ k : C n → C l by pr ¯ k : (¯ z, ¯ w ) ( π ¯ k (¯ z ) , π ¯ k ( ¯ w )) . • The coordinates of C n will be denoted by (¯ z, ¯ w, ¯ w , ¯ w ) , and Pr w willdenote the projection on the second n coordinates. • For a tuple ¯ k = ( k , . . . , k l ) ⊆ ( n ) define a map Pr ¯ k : C n → C l by Pr ¯ k : (¯ z, ¯ w, ¯ w , ¯ w ) ( π ¯ k (¯ z ) , π ¯ k ( ¯ w ) , π ¯ k ( ¯ w ) , π ¯ k ( ¯ w )) . Definition 2.2. • An algebraic variety V ⊆ C n is j -broad if for any ¯ k ⊆ ( n ) of length l wehave dim pr ¯ k V ≥ l . • An algebraic variety V ⊆ C n is j -free if pr w V is not contained in anyproper special subvariety of C n . • An algebraic variety V ⊆ C n is J -broad if for any ¯ k ⊆ ( n ) of length l wehave dim Pr ¯ k V ≥ l . • An algebraic variety V ⊆ C n is J -free if Pr w V is not contained in anyproper special subvariety of C n .2.3. The Ax-Schanuel theorem for the j -function. The Ax-Schanuel theo-rem, proven by Pila and Tsimerman, will play a key role in the proofs of the maintheorems.
VAHAGN ASLANYAN AND JONATHAN KIRBY
Fact 2.3 (Complex Ax-Schanuel for J , [PT16]) . Let V ⊆ C n be an algebraicvariety and let U be an analytic component of the intersection V ∩ Γ J . If dim U > dim V − n and no coordinate is constant on Pr w U then Pr w U is contained in aproper special subvariety of C n . We will need a uniform version of this theorem. We introduce some notationfirst. For g ∈ GL ( C ) let H g := g − H and let j g : H g → C be the function j g ( z ) = j ( gz ) . For a tuple ¯ g = ( g , . . . , g n ) ∈ GL ( C ) n let H ¯ g := H g × · · · × H g n and define the functions j ¯ g : H ¯ g → C n : ( z , . . . , z n ) ( j g ( z ) , . . . , j g n ( z n )) and J ¯ g = ( j ¯ g , j ′ ¯ g , j ′′ ¯ g ) : H ¯ g → C n : ¯ z ( j ¯ g (¯ z ) , j ′ ¯ g (¯ z ) , j ′′ ¯ g (¯ z )) where the derivation is coordinatewise and j ′ g i ( z i ) = ddz i j ( g i z i ) = j ′ ( g i z i ) · g ′ i z i , j ′′ g i ( z i ) = d dz i j ( g i z i ) . Here j ′ ( z ) = ddz j ( z ) and g ′ z = ddz ( gz ) for g ∈ GL ( C ) . We let Γ ¯ gj ⊆ H ¯ g × C n and Γ ¯ gJ ⊆ H ¯ g × C n denote the graphs of j ¯ g and J ¯ g respectively. Fact 2.4 (Uniform Ax-Schanuel for J , [Asl18b, Theorem 7.8]) . Let V ¯ s ⊆ C n bea parametric family of algebraic varieties where ¯ s ranges over a constructible set Q . Then there is a finite collection Σ of proper special subvarieties of C n suchthat for every ¯ s ∈ Q ( C ) and ¯ g ∈ GL ( C ) n , if U is an analytic component of theintersection V ¯ s ∩ Γ ¯ gJ with dim U > dim V ¯ s − n , and no coordinate is constant on Pr w U , then Pr w U is contained in some T ∈ Σ . This theorem implies the following statement for j . Fact 2.5 (Uniform Ax-Schanuel for j ) . Let ( V ¯ s ) ¯ s ∈ Q be a parametric family ofalgebraic varieties in C n . Then there is a finite collection Σ of proper specialsubvarieties of C n such that for every ¯ s ∈ Q ( C ) and every ¯ g ∈ GL ( C ) n , if U isan analytic component of the intersection V ¯ s ∩ Γ ¯ gj with dim U > dim V ¯ s − n , andno coordinate is constant on pr w U , then pr w U is contained in some T ∈ Σ . Proof of Theorem 1.8
We will prove a slightly stronger theorem. Consider the group G := (cid:18) R > R (cid:19) = (cid:26)(cid:18) a b (cid:19) : a ∈ R > , b ∈ R (cid:27) ⊆ GL +2 ( R ) . This is a two-dimensional real Lie group.
LURRINGS OF THE j -FUNCTION 7 Theorem 3.1.
Let V ⊆ H n × C n be an irreducible j -broad and j -free varietydefined over C , and let G ⊆ G be a dense subgroup (in the Euclidean topology).Then V ∩ B Gj is dense in V in the complex topology. In particular, this holds for G = GL ( Q ) ∩ G . In the proof we will work in the o-minimal structure R an , exp – the expansionof the real field by restricted analytic functions and the full exponential function– and will apply the following special case of the fibre dimension theorem in o-minimal structures. We refer the reader to [vdD98, Chapter 4, Corollary 1.6] forthe fibre dimension theorem and to [vdDMM94, vdDM94] for details on R an , exp . Fact 3.2.
Let ( R ; <, . . . ) be an o-minimal structure. Let also X ⊆ R n be a defin-able set and f : X → R m be a definable map with finite fibres. Then dim f ( X ) =dim X . In this section we will deal with real and complex dimensions of (locally) analyticsets. To distinguish between them we will use the notation dim C and dim R for thecomplex and real dimensions respectively. Proof of Theorem 3.1.
Let V ⊆ H n × C n be an irreducible j -broad and j -free va-riety. We may assume dim V = n by intersecting V with generic hyperplanes andreducing its dimension (see [Asl18a, Lemma 4.31]).Pick a tuple ¯ k = ( k , . . . , k l ) ⊆ (1 , . . . , n ) . For a tuple ¯ s ∈ pr ¯ k V ⊆ C l considerthe fibre V ¯ s ⊆ C n − l ) above ¯ s . This gives a parametric family of algebraic varieties.Let Σ ¯ k be the collection of special subvarieties of C n − l given by uniform Ax-Schanuel for this family.Further, by the fibre dimension theorem (see [Sha13, Chapter 1, §6, Theorem1.25]) there is a proper Zariski closed subset W ¯ k of pr ¯ k V such that if ¯ s / ∈ W ¯ k then(3.1) dim V ¯ s = dim V − dim pr ¯ k V ≤ n − l where the last inequality follows from the assumption that V is j -broad.Let V reg denote the set of regular points of V . It is a Zariski open subset of V .Consider the set V ′ := V reg ∩ ¯ e ∈ V : pr ¯ k ¯ e / ∈ W ¯ k , pr w pr ¯ k ¯ e / ∈ [ S ∈ Σ ¯ k S, for all ¯ k . Clearly, V ′ is a Zariski open subset of V and V ′ = ∅ as V is j -free. Hence V ′ isopen and dense in V with respect to the complex topology. Moreover, V ′ is locallyanalytic.Since V ∩ ( H n × C n ) = ∅ and dim V = n , the intersection V ′ ∩ ( H n × C n ) is a non-empty open subset of V ′ (in the complex topology) and has complex dimension n . Choose fundamental domains F , . . . , F n ⊆ H of the action of SL ( Z ) on H , VAHAGN ASLANYAN AND JONATHAN KIRBY and denote F × . . . × F n by F , so that the set V ∗ := V ′ ∩ ( F × C n ) is a non-empty (locally) analytic subset of F × C n of dimension n . Note that in generalwe may need to take different fundamental domains F , . . . , F n to ensure that V ′ ∩ ( F × C n ) = ∅ . The function j : F k → C is bijective on F k and holomorphic onthe interior F k of F k . Hence, the inverse map j − k : C → F k is well-defined on C and holomorphic on j ( F k ) .We claim that G acts transitively on H . Moreover, for any two points z , z ∈ H there is a unique element of G that maps z to z . To this end, let z = x + iy and z = u + iv where x, u ∈ R , y, v ∈ R > . Then it is straightforward to check thatthe matrix g ( z , z ) := (cid:18) vy u − xvy (cid:19) ∈ G maps z to z , and simple calculations show that it is the only element of G withthat property.Now define a map θ : F × C n → G n ,θ : (¯ z, ¯ w ) ( g ( z , j − ( w )) , . . . , g ( z n , j − n ( w n ))) , and let θ ∗ := θ | V ∗ be the restriction of θ to V ∗ .It is well known that the restriction of j to any fundamental domain is definablein the o-minimal structure R an , exp (see [PS04]). Hence, θ is a definable map. Since G and V ∗ are definable in the field structure of R , the map θ ∗ is definable in R an , exp . Claim.
All fibres of θ ∗ are finite. Proof.
The main idea of the proof is that even though θ ∗ is not a complex analyticmap (it is of course real analytic), its fibres are contained in complex analytic setsof dimension . Pick a tuple ¯ g ∈ G n . It is easy to see that θ − (¯ g ) ⊆ Γ ¯ gj ∩ ( F × C n ) . Indeed, if θ (¯ z, ¯ w ) = ¯ g then g k z k = j − k ( w k ) for all k , hence w k = j ( g k z k ) for all k. Thus, ( θ ∗ ) − (¯ g ) ⊆ V ∗ ∩ Γ ¯ gj = V ′ ∩ Γ ¯ gj ∩ ( F × C n ) . Observe that V ′ ∩ Γ ¯ gj is a locally analytic subset of H × C n . Let U be a non-empty irreducible component of it. We will show that dim C U = 0 . Pick a point ¯ e ∈ U . Note that a coordinate z k is constant on U if and only if the correspondingcoordinate w k is also constant on U , i.e. constant coordinates on U come inpairs. Let z k , . . . , z k l and the corresponding w -coordinates list all of the constantcoordinates on U . Set ¯ k := ( k , . . . , k l ) and ¯ t := pr ¯ k ¯ e , and consider the fibre U ¯ t .Then dim C U ¯ t = dim C U and U ¯ t has no constant coordinates. Moreover, U ¯ t is ananalytic component of V ′ ¯ t ∩ Γ ¯ hj where ¯ h = ( g k , . . . , g k l ) . Since ¯ e / ∈ S for any S ∈ Σ ¯ k , LURRINGS OF THE j -FUNCTION 9 we conclude by the Ax-Schanuel theorem that dim C U ¯ t = dim C V ′ ¯ t − ( n − l ) ≤ by (3.1), hence dim C U ¯ t = dim C U = 0 .Now since U is locally analytic, connected and of dimension , it must be asingleton. We showed that all analytic components of the set V ′ ∩ Γ ¯ g are singletons,hence V ′ ∩ Γ ¯ gj is discrete. So V ∗ ∩ Γ ¯ gj ⊆ V ′ ∩ Γ ¯ g is also discrete. It must actuallybe finite, for it is definable in R an , exp . (cid:3) Thus, the map θ ∗ : V ∗ → G n has finite fibres and is definable in the o-minimalstructure R an , exp . The real dimension of V ∗ is clearly n , as is the dimension of G n . Since the fibres of θ ∗ are finite, the image θ ∗ ( V ∗ ) must have dimension n byFact 3.2. In particular, θ ∗ ( V ∗ ) must contain a cell of dimension n which meansthat it has non-empty interior. Therefore, θ ∗ ( V ∗ ) has non-empty intersection withany dense subset of G n , in particular with G n .Moreover, if O ⊆ V ∗ ( C ) is an open subset (definable in R ), then dim R O =dim R V ∗ = 2 n and dim R θ ∗ ( O ) = 2 n . So θ ∗ ( O ) ⊆ G n has non-empty interior and θ ∗ ( O ) ∩ G n = ∅ . Therefore, O ∩ B Gj = ∅ . This means that V ∗ ∩ B Gj is dense in V ∗ .Since this is true for any choice of the fundamental domains, we conclude that V ∩ B Gj is dense in V . (cid:3) Remark . The proof shows that Theorem 3.1 holds for various other groups too,e.g. for dense subgroups of the subgroup of SL ( R ) consisting of upper triangularmatrices. 4. Proof of Theorem 1.7
First, we observe that the relation B GJ can be expressed in terms of the functions J ¯ g and their graphs Γ ¯ gJ (defined in Section 2.3) as follows. For a subgroup G ⊆ GL ( C ) we have B GJ = { (¯ z, J ¯ g (¯ z )) : ¯ g ∈ G n , ¯ z ∈ H ¯ g } = [ ¯ g ∈ G n Γ ¯ gJ ⊆ C n . Now let V ⊆ C n be J -broad and J -free. As in the proof of Theorem 1.8, wemay assume dim V = 3 n . We shall define a Zariski open subset V ′ of V as in theprevious section. For a tuple ¯ k = ( k , . . . , k l ) ⊆ (1 , . . . , n ) consider the parametricfamily of the fibres V ¯ s ⊆ C n − l ) for ¯ s ∈ Pr ¯ k V ⊆ C l . Let Σ ¯ k be the collectionof special subvarieties of C n − l given by uniform Ax-Schanuel with derivatives forthis family. Let also W ¯ k ⊆ Pr ¯ k V be a Zariski closed subset such that dim V ¯ s = dim V − dim Pr ¯ k V ≤ n − l ) whenever ¯ s / ∈ W ¯ k . Finally, let V ′ := V reg ∩ ¯ e ∈ V : Pr ¯ k ¯ e / ∈ W ¯ k , Pr w Pr ¯ k ¯ e / ∈ [ S ∈ Σ ¯ k S, for all ¯ k . Lemma 4.1.
Given a complex number D = 0 , there are an open and dense subset U ⊆ C (in the complex topology) and an analytic map ζ D : U → GL ( C ) suchthat for any ( z, w, w , w ) ∈ U if g = ζ D ( z, w, w , w ) then we have det g = D and (4.1) w = j ( gz ) , w = ( j ( gz )) ′ , w = ( j ( gz )) ′′ . Proof.
We solve the system of equations (4.1) with respect to g = (cid:18) a bc d (cid:19) . Ob-serve that if det g = D then ( j ( gz )) ′ = j ′ ( gz ) · ( gz ) ′ = j ′ ( gz ) · D ( cz + d ) and ( j ( gz )) ′′ = j ′′ ( gz ) · D ( cz + d ) − j ′ ( gz ) · cD ( cz + d ) . Let F ⊆ H be a fundamental domain for the action of SL ( Z ) . Then the map j − : C → F is well defined. Let ( z, w, w , w ) ∈ C and ˜ z := j − ( w ) ∈ F . Considerthe equations w = j ′ (˜ z ) · D ( cz + d ) , w = j ′′ (˜ z ) · D ( cz + d ) − j ′ (˜ z ) · cD ( cz + d ) . Solving these with respect to c and d we get c = 12 w · s Dj ′ (˜ z ) w · " j ′′ (˜ z ) · (cid:18) w j ′ (˜ z ) (cid:19) − w ,d = s Dj ′ (˜ z ) w − cz. Further, we find a and b from the equations az + bcz + d = ˜ z, ad − bc = D. The solutions for a, b, c, d are functions of z, w, w , w , and it is clear that (4.1)holds for those functions. Let F be the interior of F and let C ′ := j ( F ) . Then C ′ is open and dense in C and j − : C ′ → F is holomorphic. Choosing a branchof the square root we see that a, b, c, d are holomorphic on an open dense subset U ⊆ C × C ′ × C . (cid:3) LURRINGS OF THE j -FUNCTION 11 Now define a map θ := ( ζ , . . . , ζ ) : U n → SL ( C ) n . Consider the set V ∗ := V ′ ∩ U n and the restriction θ ∗ := θ | V ∗ . It is evident that V ∗ is a smooth analytic subset of U n of dimension n and θ ∗ is a holomorphicmap. Given a tuple of matrices ¯ g ∈ θ ∗ ( V ∗ ) , we have ( θ ∗ ) − (¯ g ) ⊆ V ′ ∩ Γ ¯ gJ . As in the proof of the claim in the previous section we can employ the Ax-Schanuel theorem to show that V ′ ∩ Γ ¯ gJ is discrete. Thus, θ ∗ has discrete fibres.Since dim C V ∗ = dim C SL ( C ) n = 3 n , by Remmert’s open mapping theorem (see[Łoj91, §V.6, Theorem 2]) we conclude that θ ∗ is an open map. Therefore forany open subset O ⊆ V ∗ the image θ ∗ ( O ) ⊆ SL ( C ) n is open, so it intersects G n .Thus, any open subset of V ∗ intersects B GJ and so V ∩ B GJ is dense in V .Finally, if C is a subfield of C not contained in R , then C is dense in C and SL ( C ) is dense in SL ( C ) and the second part of Theorem 1.7 follows.We can deduce Theorem 1.4 from Theorem 1.7. Proof of Theorem 1.4. If V ⊆ C n is j -broad and j -free then ˜ V := V × C n ⊆ C n is J -broad and J -free, hence ˜ V ( C ) ∩ B GJ is dense in ˜ V ( C ) . Hence, V ∩ B Gj is densein V . (cid:3) Model theoretic properties of the blurred j -function In this section we show that for a suitably chosen countable algebraically closedsubfield C ⊆ C and the group G := GL ( C ) the structures C B Gj := ( C ; + , · , B Gj ) and C B GJ := ( C ; + , · , B GJ ) are model theoretically tame, where B Gj and B GJ areconsidered respectively as binary and -ary relations. In particular, we get a first-order axiomatisation of those structures and show that they are ω -stable of Morleyrank ω .5.1. j -reducts of differentially closed fields. The j -function satisfies a thirdorder algebraic differential equation over Q . Namely, Ψ( j, j ′ , j ′′ , j ′′′ ) = 0 where Ψ( w, w , w , w ) = w w − (cid:18) w w (cid:19) + w − w + 26542082 w ( w − · w . Thus Ψ( w, w ′ , w ′′ , w ′′′ ) = Sw + R ( w )( w ′ ) , where S denotes the Schwarzian derivative defined by Sw = w ′′′ w ′ − (cid:18) w ′′ w ′ (cid:19) and R ( w ) = w − w + 26542082 w ( w − . Note that all functions j ( gz ) with g ∈ GL ( C ) satisfy the differential equation Ψ( w, w ′ , w ′′ , w ′′′ ) = 0 and in fact all solutions are of that form (see [FS15, Lemma4.2] or [Asl18a, Lemma 4.1]).In a differential field ( K ; + , · , ′ ) for a non-constant z ∈ K define a derivation ∂ z : K → K by ∂ z : w w ′ z ′ . Consider the equation χ ( z, w ) := Ψ( w, ∂ z w, ∂ z w, ∂ z w ) = 0 . Let E j ⊆ K be a binary relation interpreted as the set of solutions of the equation χ ( z, w ) = 0 (multiplied through by a common denominator to make χ into adifferential polynomial). The reduct ( K ; + , · , E j ) will be denoted by K E j . Fact 5.1 ([Asl18a, Theorem 4.39] and [AEK20, Theorem 1.1]) . Let ( K ; + , · , D ) be a differentially closed field with field of constants C . Then T j := Th( K E j ) isaxiomatised by the following axioms and axiom schemes. A1 K is an algebraically closed field of characteristic . A2 C := { c ∈ K : E j (0 , c ) } is an algebraically closed subfield. Further, C ⊆ E j ( K ) and if ( z, j ) ∈ E j ( K ) and one of z, j is constant then both of themare constants. A3 If ( z, w ) ∈ E j then for any g ∈ SL ( C ) , ( gz, w ) ∈ E j . Conversely, if forsome w we have ( z , w ) , ( z , w ) ∈ E j then z = gz for some g ∈ SL ( C ) . A4 If ( z, w ) ∈ E j and Φ N ( w , w ) = 0 for some w and some modular poly-nomial Φ N ( X, Y ) then ( z, w ) ∈ E j . AS If ( z i , w i ) ∈ E j , i = 1 , . . . , n, with td C C (¯ z, ¯ w ) ≤ n, then Φ N ( w i , w k ) = 0 for some N and some ≤ i < k ≤ n , or w i ∈ C forsome i . EC For each j -broad variety V ⊆ K n the intersection E j ( K ) ∩ V ( K ) is non-empty. NT There is a non-constant element in K .Furthermore, T j is ω -stable of Morley rank ω , and near model complete, i.e. ev-ery formula is equivalent to a Boolean combination of existential formulas modulo T j . LURRINGS OF THE j -FUNCTION 13 Here AS should be understood as the uniform Ax-Schanuel which is first-orderexpressible (see [Asl18a, p. 26] and [Asl18b, Theorem 4.6]).Similarly, if we let E J be the set of all -tuples ( w, ∂ z w, ∂ z w, ∂ z w ) ∈ K with χ ( z, w ) = 0 then the first-order theory of K E J := ( K ; + , · , E J ) is well understood.In particular, a complete axiomatisation has been obtained in [Asl18a, Theorem5.20].5.2. First-order theory of the blurred j -function.Definition 5.2 ([Ete18, §5]) . A j -derivation on the field of complex numbers is aderivation δ : C → C such that for any z ∈ H we have δj ( z ) = j ′ ( z ) δ ( z ) , δj ′ ( z ) = j ′′ ( z ) δ ( z ) , δj ′′ ( z ) = j ′′′ ( z ) δ ( z ) . The space of j -derivations is denoted by j Der( C ) .Let C := \ δ ∈ j Der( C ) ker δ. Then C is a countable algebraically closed subfield of C and j ( C ∩ H ) = C (see[Ete18, §5]). Theorem 5.3 (cf. [Kir19, Theorem 1.4]) . Let C be as above and G = GL ( C ) .Then Th( C B Gj ) = T j . In particular, Th( C B Gj ) is ω -stable with Morley rank ω andis near model complete.Proof. Since T j is complete, it suffices to prove that C B Gj | = T j . Axioms A1-A4and NT are straightforward to check, AS follows from [Ete18, Proposition 6.2] andEC follows from Theorem 1.8, for it is proven in [Asl18a, §4.8] that in EC one mayassume the variety is free. (cid:3) Similarly, we get the following result for the blurred J -function. Theorem 5.4.
Let C be as above and G = GL ( C ) . Then C B GJ is elemen-tarily equivalent to E J -reducts of differentially closed fields, axiomatised by A1 ′ -A4 ′ ,AS ′ ,EC ′ ,NT ′ from [Asl18a, §5]. In particular, Th( C B GJ ) is ω -stable with Morleyrank ω and is near model complete. Proof.
This can be proven exactly as Theorem 5.3. (cid:3)
Remarks on quasiminimality.
The structures C B Gj and C B GJ are quasimin-imal , that is, every first-order definable set in either of those structures is countableor its complement is countable. Let us explain how this can be proven withoutgoing into the details. We focus on C B GJ , the case of C B Gj being analogous. Thereare three main ingredients in the proof. (a) There is a natural pregeometry on C B GJ given by the Ax-Schanuel inequal-ity, denoted by cl . See [Asl18a, Definition 2.15].(b) C B GJ has the countable closure property , i.e. the cl -closure of a finite set iscountable.(c) C B GJ realises a unique generic type over countable closed subsets: if A is acountable cl -closed subset and a , a ∈ C \ A then tp( a /A ) = tp( a /A ) .For details on (a) we refer the reader to [Asl18a], (b) follows from the proof ofTheorem 1.7, and (c) can be proven by constructing a back-and-forth system ofpartial isomorphisms from a to a over A as in the proof of [Asl18a, Proposition4.38] (and it relies on a “generic” version of Theorem 1.7). Now, given a set X ⊆ C ,definable in C B GJ , let A be the closure of the finitely many parameters used in thedefinition of X . Then A is countable and by (c) either X ⊆ A or X ⊇ C \ A .In [Kir19] Kirby proves a stronger quasiminimality result for the blurred complexexponentiation, namely, every subset which is invariant under all automorphismsis either countable or co-countable. It is likely that the methods of [Kir19] will gothrough for the j -function and we will have similar stronger results for C B Gj and C B GJ .Unlike complex exponentiation, where quasiminimality is an open question, itis clear that the j -function itself cannot be quasiminimal. However, it would beinteresting to understand which blurrings of the j -function are quasiminimal. Sincethe action of GL ( C ) factors through PGL ( C ) , we ask the following question. Question 5.5.
For which proper subgroups G of PGL ( C ) are the structures C B Gj and C B GJ quasiminimal? One can see immediately that there are some trivial examples where C B Gj is notquasiminimal. First, when G is uncountable, the fibres of B Gj above the secondcoordinate are uncountable with an uncountable complement, hence G must beat most countable. Second, if G ⊆ PGL ( R ) , then the projection of B Gj on thefirst coordinate is H , therefore C B Gj is not quasiminimal. Further, when G is finitethen the fibres of B Gj above the first coordinate may be finite and of differentcardinalities which allows one to define an uncountable set whose complement isalso uncountable. For example, when G is the group generated by (cid:18) i
00 1 (cid:19) , thenthe set S := { z ∈ C : ∃ ! w B Gj ( z, w ) } contains R ∪ i R \{ } , so it is uncountable andone can easily see that S \ ( R ∪ i R ) is at most countable, hence C \ S is uncountable.It seems plausible that the above question has an affirmative answer if and onlyif G * PGL ( R ) and G is countably infinite. LURRINGS OF THE j -FUNCTION 15 References [AEK20] Vahagn Aslanyan, Sebastian Eterović, and Jonathan Kirby. Differential ExistentialClosedness for the j -function. Preprint, arXiv:2003.10996, 2020.[Asl18a] Vahagn Aslanyan. Adequate predimension inequalities in differential fields.Preprint, arXiv:1803.04753, 2018.[Asl18b] Vahagn Aslanyan. Weak Modular Zilber-Pink with Derivatives. Preprint,arXiv:1803.05895, 2018.[BM17] Dale Brownawell and David Masser. Zero estimates with moving targets. J. Lond.Math. Soc. , 95(2):441–454, 2017.[DFT18] Paola D’Aquino, Antongiulio Fornasiero, and Giuseppina Terzo. Generic solutionsof equations with iterated exponentials.
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Vahagn Aslanyan, School of Mathematics, University of East Anglia, Norwich,NR4 7TJ, UK
E-mail address : [email protected]
Jonathan Kirby, School of Mathematics, University of East Anglia, Norwich,NR4 7TJ, UK
E-mail address ::