Differential Existential Closedness for the j -function
aa r X i v : . [ m a t h . L O ] S e p DIFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION VAHAGN ASLANYAN, SEBASTIAN ETEROVIĆ, AND JONATHAN KIRBY
Abstract.
We prove the Existential Closedness conjecture for the differen-tial equation of the j -function and its derivatives. It states that in a differ-entially closed field certain equations involving the differential equation of the j -function have solutions. Its consequences include a complete axiomatisation of j -reducts of differentially closed fields, a dichotomy result for strongly minimalsets in those reducts, and a functional analogue of the Modular Zilber-Pink withDerivatives conjecture. Introduction
Let H be the complex upper half-plane and j : H → C be the modular j -function. The transcendence properties of this function have been well studied.For example, it is well known that for a matrix g ∈ GL ( C ) the functions j ( z ) and j ( gz ) are algebraically dependent if and only if g has rational entries (and positivedeterminant), and in that case the dependence is given by a modular polynomial .It is also known that j ( z ) , j ′ ( z ) , j ′′ ( z ) are algebraically independent functions over C ( z ) and j ′′′ ∈ Q ( j, j ′ , j ′′ ) [Mah69]. These statements are generalised by a theoremof Pila and Tsimerman [PT16], known as Ax-Schanuel for the j -function, for it isan analogue of the Ax-Schanuel theorem for the exponential function [Ax71]. Aprecise statement of Ax-Schanuel will be given in Section 2, but we give a roughdescription now. The j -function satisfies an order three differential equation (seeSection 2), and the Ax-Schanuel theorem is a transcendence statement about itssolutions often formulated in a differential algebraic language. It can be seen as anecessary condition for a system of differential equations in terms of the equationof the j -function to have a solution in a differential field F . In particular, it impliesthat “overdetermined” systems cannot have a solution. Date : September 8, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Ax-Schanuel, Existential Closedness, j -function.VA and JK were supported by EPSRC grant EP/S017313/1. SE was partially supported byNSF RTG grant DMS-1646385. Let E j ( x, y ) denote the differential equation of the j -function. Here we think of x as some complex function and of y as j ( x ) , although a solution to the differentialequation could be any function j ( gx ) for any g ∈ GL ( C ) . Solving a system ofequations is equivalent to finding points of the form (¯ z, ¯ j ) in an algebraic variety V ⊆ F n such that each ( z i , j i ) is in E j . The notion of an overdetermined systemcan be captured by some dimension conditions on the algebraic variety V ⊆ F n dictated by the Ax-Schanuel theorem, and varieties not satisfying those conditionsare called j -broad . j -broadness of V means that V is not too small, in particular,it implies that dim V ≥ n . See Section 3.5 for a definition.Thus, in some sense Ax-Schanuel states that if V contains a “sufficiently generic” E j -point, then V must be j -broad. There is a dual statement according to which j -broad varieties do contain E j -points in suitably large differential fields, namely,differentially closed fields. This dual statement is known as Existential Closedness ,henceforth referred to as EC, and was conjectured in [Asl18a]. In this paper weprove it and establish several related results. The following is one of the maintheorems of the paper.
Theorem 1.1.
Let F be a differential field, and V ⊆ F n be a j -broad variety.Then there is a differential field extension K ⊇ F such that V ( K ) contains an E j -point. In particular, if F is differentially closed then V ( F ) contains an E j -point. We actually prove a more general theorem incorporating the j -function and itsfirst two derivatives. Let E J ( x, y, y ′ , y ′′ ) be a -ary relation where, by definition, ( x, y ) ∈ E j and we think of y ′ and y ′′ as the first and second derivatives of y with respect to x . In other words, we may think of x as a complex function,and of y, y ′ , y ′′ as j ( x ) , j ′ ( x ) , j ′′ ( x ) where j ′ , j ′′ are the first two derivatives of the j -function. We do not consider further derivatives since j ′′′ is already algebraicover j, j ′ , j ′′ . The Ax-Schanuel theorem is in fact a transcendence statement for E J -points in a differential field. As above, we define J - broad varieties V ⊆ F n bysome dimension conditions dictated by Ax-Schanuel. In particular, if V ⊆ F n is J -broad then dim V ≥ n + 1 . See Section 3.1 for a precise definition. Then wehave the following EC statement for E J which was conjectured in [Asl18a]. Structures satisfying EC may not be existentially closed (in a first-order sense) in arbitraryextensions, but they are existentially closed in so called “strong” extensions. See [Asl18a, Kir09]for details. In this paper all fields are of characteristic . The use of upper case J in some notations and terms is motivated by the fact that the vectorfunction ( j, j ′ , j ′′ ) : H → C is often denoted by J . Note that both j -broadness and J -broadness are algebraic (as opposed to differential) prop-erties of a variety V and they just state that certain projections of V have sufficiently largedimension. IFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION 3 Theorem 1.2.
Let F be a differential field, and V ⊆ F n be a J -broad variety.Then there is a differential field extension K ⊇ F such that V ( K ) contains an E J -point. In particular, if F is differentially closed then V ( F ) contains an E J -point. Further, we prove that certain varieties contain non-constant E J -points. Thosevarieties are called strongly J -broad . Such a variety (in F n ) must have dimension ≥ n + 1 . Theorem 1.3.
Let F be a differentially closed field, and let V ⊆ F n be a strongly J -broad variety defined over the field of constants C . Then there is a differentialfield extension K ⊇ F such that V ( K ) contains an E j -point none of the coordinatesof which is constant in K . In particular, if F is differentially closed then V ( F ) contains an E J -point with no constant coordinates. As mentioned above, the original Ax-Schanuel theorem is about the exponen-tial differential equation and the appropriate EC statement in that setting wasestablished by Kirby in [Kir09]. However, he uses the proof of Ax’s theorem whichis differential algebraic and allows one to work with an abstract differential fieldand differential forms. On the other hand, there is no known differential algebraicproof for the Ax-Schanuel theorem for the j -function, hence we cannot use its proof(the proof of [PT16] uses o-minimality and point counting). Instead we use thestatement of the Ax-Schanuel theorem itself, regarding it as a black box withoutlooking inside it. This method is quite general and would probably adapt to othersettings to prove an appropriate EC statement as long as there is an appropriateAx-Schanuel statement. To demonstrate this, in Section 4 we give a proof of ECfor the exponential differential equation. In fact, we prove EC for the exponentialdifferential equation in fields with several commuting derivations, which is techni-cally a new result as it is not addressed in Kirby’s work. An analogous result forthe j -function holds as well, with a similar proof.Note that Kirby’s work on EC for the exponential differential equation, as wellas the current paper on EC for the differential equation of the j -function, havebeen motivated by Zilber’s work on pseudo-exponential fields and, in particular, his Exponential Closedness conjecture for the complex exponential function (see [Zil04,BK18]). Indeed, the differential EC for exp is a functional/differential analogue ofZilber’s conjecture, and the differential EC for j is again its functional/differentialanalogue for the j -function. Naturally, there is a complex existential closednessconjecture for the j -function which states roughly that j -broad varieties containpoints of the form (¯ z, j (¯ z )) where each z i is a complex number in the upper half-plane, and j is the j -function. There is also a similar conjecture for j and itsderivatives. Both of those are analogues of Zilber’s conjecture mentioned above.Eterović and Herrero have recently made progress towards the complex EC for the j -function [EH20]. VAHAGN ASLANYAN, SEBASTIAN ETEROVIĆ, AND JONATHAN KIRBY
We remark that the EC theorem for the j -function has several important conse-quences. In particular, in [Asl18a, §4] Aslanyan studied E j -reducts of differentiallyclosed fields, that is, reducts of the form ( K ; + , · , E j ) where E j is a binary pred-icate for the solutions of the differential equation of the j -function and K is adifferentially closed field with a single derivation. He gave a candidate axioma-tisation of the common complete theory of those reducts, and proving EC showsthat it is indeed an axiomatisation of that theory. It can be thought of as thefirst-order theory of the differential equation of the j -function. EC for j and itsderivatives gives a similar axiomatisation of the E J -reducts of differentially closedfields (see [Asl18a, §5]). In the terminology of [Asl18a] these results mean that theAx-Schanuel inequality for E J (as well as E j ) is adequate .Using the methods and results of [Asl18a] and assuming EC for E j , Aslanyan alsoproved a dichotomy result for strongly minimal sets in E j -reducts of differentiallyclosed fields. That statement now becomes unconditional due to this paper (see[Asl20]).Further, EC (with derivatives) was used in [Asl18b] to establish a Zilber-Pinktype statement for the j -function and its derivatives. Note that Zilber-Pink is adiophantine conjecture generalising Mordell-Lang and André-Oort (see, for exam-ple, [Zan12, HP16, DR18] for details). Two such statements were considered in[Asl18b] one of which was proven unconditionally and the other was proven condi-tionally upon the EC conjecture, so proving the latter here makes it unconditional.The current paper may be seen as a continuation of [Asl18a]. Nevertheless, theresults of this paper, though motivated by [Asl18a], are quite independent fromthe latter. We have tried to make this paper as self-contained as possible. Mostproofs are based on basic facts from differential algebra and algebraic geometry,and of course the Ax-Schanuel theorem.2. Ax-Schanuel for the j -function Modular polynomials.
The j -function is a modular function defined andholomorphic on the upper half-plane H := { z ∈ C : Im z > } . It is invariantunder the action of SL ( Z ) on H , and satisfies certain algebraic “functional equa-tions” under the action of GL +2 ( Q ) – the group of × rational matrices withpositive determinant. More precisely, there is a countable collection of irreduciblepolynomials 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 ) . Two elements w , w ∈ C are called modularly independent if they do not satisfyany modular relation Φ N ( w , w ) = 0 . We refer the reader to [Lan73] for furtherdetails. IFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION 5 Differential equation.
The j -function satisfies an order algebraic differen-tial equation over Q , and none of lower order (see [Mah69]). Namely, Ψ( j, j ′ , j ′′ , j ′′′ ) =0 where Ψ( y , y , y , y ) = y y − (cid:18) y y (cid:19) + y − y + 26542082 y ( y − · y . Thus Ψ( y, y ′ , y ′′ , y ′′′ ) = Sy + R ( y )( y ′ ) , where S denotes the Schwarzian derivative defined by Sy = y ′′′ y ′ − (cid:16) y ′′ y ′ (cid:17) and R ( y ) = y − y +26542082 y ( y − is a rational function.It is well known that all functions j ( gz ) with g ∈ SL ( C ) satisfy the differentialequation Ψ( y, y ′ , y ′′ , y ′′′ ) = 0 and all solutions (not necessarily defined on H ) areof that form (see [FS18, Asl18a]).Note that for non-constant y , the relation Ψ( y, y ′ , y ′′ , y ′′′ ) = 0 is equivalent to y ′′′ = η ( y, y ′ , y ′′ ) where η ( y, y ′ , y ′′ ) := 32 · ( y ′′ ) y ′ − R ( y ) · ( y ′ ) is a rational function over Q . Ax-Schanuel.
From now on, y ′ , y ′′ , y ′′′ will denote some variables/coordinatesand not the derivatives of y . Derivations of abstract differential fields will not bedenoted by ′ . Definition 2.1.
Let ( F ; + , · , D , . . . , D m ) be a differential field with constant field C = T mk =1 ker D k . We define a -ary relation E J ( x, y, y ′ , y ′′ ) by ∃ y ′′′ " Ψ( y, y ′ , y ′′ , y ′′′ ) = 0 ∧ m ^ k =1 D k y = y ′ D k x ∧ D k y ′ = y ′′ D k x ∧ D k y ′′ = y ′′′ D k x . The relation E × J ( x, y, y ′ , y ′′ ) is defined by the formula E J ( x, y, y ′ , y ′′ ) ∧ x / ∈ C ∧ y / ∈ C ∧ y ′ / ∈ C ∧ y ′′ / ∈ C. Pila and Tsimerman have established the following transcendence result for the j -function which is an analogue of Ax’s theorem for the exponential function (see[Ax71], and also Section 4 for a statement). It will play a crucial role in this paper. Fact 2.2 (Ax-Schanuel for j , [PT16]) . Let ( F ; + , · , D , . . . , D m ) be a differen-tial field with commuting derivations and with field of constants C . Let also ( z i , j i , j ′ i , j ′′ i ) ∈ E × J ( F ) , i = 1 , . . . , n. If the j i ’s are pairwise modularly indepen-dent then td C C (¯ z, ¯ j, ¯ j ′ , ¯ j ′′ ) ≥ n + rk( D k z i ) i,k . VAHAGN ASLANYAN, SEBASTIAN ETEROVIĆ, AND JONATHAN KIRBY
As pointed out in the introduction, this statement can be interpreted as follows.Given a variety V ⊆ F n , if dim V or the dimension of some projections of V istoo small, then V ( F ) cannot contain an E × J ( F ) -point whose j -coordinates (i.e.the second n coordinates) are pairwise modularly independent. The definition of J -broadness given in the next section is based on this observation. ExistentialClosedness, which is the main result of this paper, is a dual to Ax-Schanuel. Itstates that if V is J -broad then it contains an E J -point.Note that we will actually work with varieties defined over arbitrary (not nec-essarily constant) parameters, and we will look for E J -points in those varieties,rather than E × J -points. We also prove a theorem about certain varieties contain-ing E × J -points (Theorem 3.6). It may seem more appropriate to think of thatstatement as the dual of Ax-Schanuel, however it is weaker than the full EC andthe latter is in fact a dual to a “relative predimension inequality” governed byAx-Schanuel. We refer the reader to [Asl18a] for details.3. Existential Closedness J -broad and J -free varieties.Definition 3.1. Let n be a positive integer, k ≤ n and ¯ i := ( i , . . . , i k ) with ≤ i < . . . < i k ≤ n . Define the projection map pr ¯ i : K n → K k by pr ¯ i : ( x , . . . , x n ) ( x i , . . . , x i k ) . Also define Pr ¯ i : K n → K k by Pr ¯ i : (¯ x, ¯ y, ¯ y ′ , ¯ y ′′ ) (pr ¯ i ¯ x, pr ¯ i ¯ y, pr ¯ i ¯ y ′ , pr ¯ i ¯ y ′′ ) . Definition 3.2.
Let K be an algebraically closed field. An irreducible algebraicvariety V ⊆ K n is J -broad if for any ≤ i < . . . < i k ≤ n we have dim Pr ¯ i ( V ) ≥ k . We say V is strongly J -broad if the strict inequality dim Pr ¯ i ( V ) > k holds. Definition 3.3.
An algebraic variety V ⊆ K n (with coordinates (¯ x, ¯ y, ¯ y ′ , ¯ y ′′ ) ) is J - free if no coordinate is constant on V , and it is not contained in any varietydefined by an equation Φ N ( y i , y k ) = 0 for some modular polynomial Φ N and someindices i, k .3.2. EC for varieties defined over arbitrary parameters.Lemma 3.4.
Let ( F ; + , · , D ) be a differential field and ( z, j, j ′ , j ′′ ) ∈ E J ( F ) with j = 0 , , j ′ , j ′′ = 0 . If one of the coordinates of ( z, j, j ′ , j ′′ ) is constant then soare the others.Proof. If Dz = 0 then from the definition of E J we get Dj = Dj ′ = Dj ′′ = 0 . If Dj = 0 or Dj ′ = 0 then since j ′ , j ′′ = 0 , we must have Dz = 0 . IFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION 7 Now assume Dj ′′ = 0 and Dz = 0 . Then η ( j, j ′ , j ′′ ) = 0 . For simplicity we willdenote j ′′ by c . Then j ′ = cz + d for some constant d , and j = c · z + d · z + e for some constant e . Substituting this into η ( j, j ′ , j ′′ ) = 0 we get a non-trivialequation for z over c, d, e , which implies that z must be constant. (cid:3) Theorem 3.5.
Let ( F ; + , · , D ) be a differential field, and let V ⊆ F n be a J -broadand J -free variety. Then there is a differential field extension K ⊇ F such that V ( K ) ∩ E J ( K ) = ∅ .Proof. This is inspired by the proof of [Kir09, Theorem 3.10]. An important differ-ence is that instead of using Proposition 3.7 of that paper, which is an intermediatestatement in the proof of Ax-Schanuel, we use Ax-Schanuel itself.Extending F if necessary we may assume that D = 0 . Let ¯ v := (¯ z, ¯ j, ¯ j ′ , ¯ j ′′ ) bea generic point of V over F , and let K := F (¯ v ) alg (we can think of ¯ v as a pointdefined in a transcendental extension of F ). We will show that we can extend D toa derivation on K such that ¯ v ∈ E J ( K ) . If dim V = 3 n + l then td( K/F ) = 3 n + l (where l ≥ ).Let Der(
K/C ) be the K -vector space of all derivations δ : K → K which vanishon C . Consider the subspace Der(
K/D ) := { δ ∈ Der(
K/C ) : δ | F = λD for some λ ∈ K } . For every δ ∈ Der(
K/D ) there is a unique λ ∈ K such that δ | F = λD , and wedenote it by λ δ . The map ϕ : δ λ δ is a linear map ϕ : Der( K/D ) → K . It issurjective, since for every λ ∈ K the map λD can be extended to a derivation of K over C . Moreover, ker( ϕ ) = Der( K/F ) , hence dim Der( K/D ) = dim Der(
K/F ) + 1 = td(
K/F ) + 1 = 3 n + l + 1 . Consider the sequence of inclusions
Der(
K/F ) ֒ → Der(
K/D ) ֒ → Der(
K/C ) . The space
Der(
K/C ) can be identified with the dual space of Ω( K/C ) – the vectorspace of differential -forms on K over C – which gives a sequence of surjections Ω( K/C ) ։ Ω( K/D ) ։ Ω( K/F ) . More precisely, Ω( K/C ) is the vector space generated by the set of symbols { dx : x ∈ K } modulo the relations d ( x + y ) = dx + dy, d ( xy ) = xdy + ydx, dc = 0 , c ∈ C. The map d : K → Ω( K/C ) is called the universal derivation on K over C (see also [Asl18b, §2]). Then Ω( K/D ) can be defined as the dual of Der(
K/D ) . It can also be defined as a quotient of Ω( K/C ) so that Der(
K/D ) is identified with the dual of Ω( K/D ) . VAHAGN ASLANYAN, SEBASTIAN ETEROVIĆ, AND JONATHAN KIRBY
This allows us to consider elements of Ω( K/C ) as elements of Ω( K/F ) by identi-fying them with their images. The following equalities are straightforward: dim Der( K/F ) = dim Ω(
K/F ) = td(
K/F ) = 3 n + l, dim Ω( K/D ) = dim Der(
K/D ) = 3 n + l + 1 . Consider the following differential forms in Ω( K/C ) : ω i := dj i − j ′ i dz i , ω ′ i := dj ′ i − j ′′ i dz i , ω ′′ i := dj ′′ i − η ( j i , j ′ i , j ′′ i ) dz i . Note that since we assumed no coordinate is constant on V , none of the coordinatesof ¯ v can be in F . Therefore η ( j i , j ′ i , j ′′ i ) is well defined. Set Λ( K/C ) := span K { ω i , ω ′ i , ω ′′ i : i = 1 , . . . , n } ⊆ Ω( K/C ) and let Λ( K/D ) ⊆ Ω( K/D ) and Λ( K/F ) ⊆ Ω( K/F ) be the images of Λ under the canonical surjec-tions. Claim.
The forms ω i , ω ′ i , ω ′′ i , i = 1 , . . . , n are K -linearly independent in Ω( K/F ) ,that is, dim Λ( K/F ) = 3 n . Proof.
We proceed by contradiction, so assume dim Λ(
K/F ) < n . Consider theannihilator Ann(Λ(
K/F )) ⊆ Der(
K/F ) . Clearly, r := dim Ann(Λ( K/F )) = dim Ω(
K/F ) − dim Λ( K/F ) > l. It is easy to see that
Ann(Λ(
K/F )) is closed under the Lie bracket, hence we canchoose a commuting basis of derivations D , . . . , D r ∈ Ann(Λ(
K/F )) (see [Kol85,Chapter 0, § , Proposition 6] or [Sin07, Lemma 2.2]). Let L := T ri =1 ker D i be thefield of constants. Then F ⊆ L ( K .Since r > l ≥ , at least one of the coordinates of ¯ v is not in L . Let v i :=( z i , j i , j ′ i , j ′′ i ) . By Lemma 3.4 we may assume that for some t ≥ none of thecoordinates of v , . . . , v t is in L and all coordinates of v t +1 , . . . , v n are in L . Let ¯ u := ( v , . . . , v t ) , ¯ w := ( v t +1 , . . . , v n ) . Since ¯ v is generic in V over F , and V is J -free, j , . . . , j t are pairwise modularlyindependent. By the Ax-Schanuel theorem for several commuting derivations td( L (¯ u ) /L ) ≥ t + rk( D i z k ) ≤ k ≤ t, ≤ i ≤ r = 3 t + r. Here we used the fact that rk( D i z k ) ≤ k ≤ t, ≤ i ≤ r = r . To show this we observe thatthe Jacobian of ¯ v with respect to D , . . . , D r has rank r (see, for example, [Asl18b, § . , Claim 1]) and for k > t all coordinates of v k are in L and hence do notcontribute to the rank of the Jacobian. Furthermore, the rows of the Jacobianof ¯ v corresponding to j i , j ′ i , j ′′ i are linearly dependent on the row correspondingto z i , so those rows do not contribute to the rank either. Thus, rk Jac(¯ v ) =rk( D i z k ) ≤ k ≤ t, ≤ i ≤ r . This is the only place in the proof where we use the fact that no coordinate is constant on V . So in fact assuming y i = 0 , and y ′ i = 0 on V would suffice. IFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION 9 Further, td(
L/F ) ≥ td( F ( ¯ w ) /F ) ≥ n − t ) for V is J -broad. Combining these two inequalities we get td( K/F ) = td(
K/L ) + td(
L/F ) ≥ t + r + 3( n − t ) = 3 n + r > n + l, which is a contradiction. (cid:3) Now we have dim Ann Λ(
K/D ) = dim Ω(
K/D ) − dim Λ( K/D ) = 3 n + l +1 − n = l +1 and dim Ann Λ( K/F ) = dim Ω(
K/F ) − dim Λ( K/F ) = l . Choose a derivation δ ∈ Ann Λ(
K/D ) \ Ann Λ(
K/F ) . Then δ | F = λ δ · D for some λ ∈ K . On the otherhand, δ / ∈ Ann(Λ(
K/F )) , therefore δ | F = 0 and λ δ = 0 . Replacing δ by λ − δ · δ wemay assume that λ δ = 1 and δ is an extension of D to K . (cid:3) Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2.
If the field F is differentially closed, then it is existentiallyclosed in K , that is, a system of differential equations with parameters from F has a solution in K if and only if it has a solution in F . Therefore, in this case V ( F ) ∩ E J ( F ) = ∅ .Now we show that in Theorem 3.5 the assumption of J -freeness may be dropped(cf. [Asl18a, Lemmas 4.43 and 5.24]). We proceed by induction on n , the case n = 1 being trivial. Let V ⊆ F n be a J -broad variety. If V is J -free, then we aredone by Theorem 3.5. So we assume V is not J -free.If V has a constant coordinate, then assume without loss of generality thatone of the coordinates x , y , y ′ , y ′′ , say x , is constant, and denote it by a . By J -broadness, dim Pr V = 3 . Choose elements b, b ′ , b ′′ in an extension F of F with ( a, b, b ′ , b ′′ ) ∈ E J ( F ) and let W ⊆ F n − be the fibre of V above ( a, b, b ′ , b ′′ ) .Then, by the fibre dimension theorem, W is J -broad so by the induction hypothesisit contains an E J -point.Now assume a modular relation Φ N ( y , y ) = 0 holds on V . Choose genericconstants a, b, c, d in an extension F ⊇ F and consider the variety S ⊆ F n givenby the equations Φ N ( y , y ) = 0 , ax + bcx + d = x and two more equations obtainedby differentiating Φ N ( y , y ) = 0 (see [Asl18b, § . ]). Let ¯ i := (2 , . . . , n ) andlet W := Pr ¯ i ( V ∩ S ) ⊆ F n − . We claim that W is J -broad. To this end let ¯ l := ( l , . . . , l m ) with ≤ l < . . . < l m ≤ n , and let ˆ l := (1 , l , . . . , l m ) . Since V is J -broad, dim Pr ˆ l V ≥ m + 1) . Since on S the coordinates x , y , y ′ , y ′′ are algebraically related to the coordinates x , y , y ′ , y ′′ respectively and there areno relations between any other coordinates, dim Pr ˆ l S = dim Pr ¯ l S = 4 m , andsimilarly dim Pr ¯ l ( V ∩ S ) = dim Pr ˆ l ( V ∩ S ) = dim(Pr ˆ l V ∩ Pr ˆ l S ) . Using the theorem on dimension of intersection of two varieties (see [Sha13, §1.6.2,Theorem 1.24]) and taking into account the fact that the relation Φ N ( y , y ) = 0 holds both on V and S , we conclude that dim Pr ¯ l W = dim(Pr ˆ l V ∩ Pr ˆ l S ) ≥ dim Pr ˆ l V + 4 m + 1 − m + 1) ≥ m. So by the induction hypothesis W contains an E J -point in a differential fieldextension of F . Then the equations defining S allow us to extend it to an E J -point of V . (cid:3) EC for varieties defined over the constants.Theorem 3.6.
Let ( F ; + , · , D ) be a differential field, and let V ⊆ F n be a strongly J -broad and J -free variety defined over the field of constants C . Then there existsa differential field extension K of F such that V ( K ) ∩ E × J ( K ) = ∅ . In particular,when F is differentially closed, we have V ( F ) ∩ E × J ( F ) = ∅ . This follows from EC by the method of intersecting V with generic hyperplanes(see [Asl18a, Lemma 4.31]). Still, we give a direct proof below, especially as itplays a key role in the proof of a Zilber-Pink type theorem in [Asl18b]. Proof.
We will make use of vector spaces of derivations and differential formsdefined in the proof of Theorem 3.5.Let ¯ v := (¯ z, ¯ j, ¯ j ′ , ¯ j ′′ ) be a generic point of V over C , and let L := C (¯ v ) alg . If dim V = 3 n + l then td( L/C ) = 3 n + l (where l ≥ ). So r := dim Ann Λ( L/C ) = dim Ω(
L/C ) − dim Λ( L/C ) ≥ l. Note that we actually know that dim Λ(
L/C ) = 3 n and the above inequality isactually an equality, but we do not need it. Pick a commuting basis of derivations D , . . . , D r of Ann Λ(
L/C ) . Consider the differential field ( L ; + , · , D , . . . , D r ) andlet C L := T ri =1 ker D i be its constant field.We claim that none of the coordinates of ¯ v is in C L . To this end assume byLemma 3.4 that for some ≤ t < n none of the coordinates of v , . . . , v t is in C L and all coordinates of v t +1 , . . . , v n are in C L , where v i := ( z i , j i , j ′ i , j ′′ i ) . Let ¯ u := ( v , . . . , v t ) , ¯ w := ( v t +1 , . . . , v n ) . Since V is strongly J -broad, td( C L /C ) ≥ td( C ( ¯ w ) /C ) ≥ n − t ) + 1 . Further, by Ax-Schanuel td( C L (¯ u ) /C L ) ≥ t + r .Combining these inequalities we get td( L/C ) = td(
L/C L ) + td( C L /C ) ≥ n + r + 1 ≥ n + l + 1 which is a contradiction. Thus, t = n and ¯ v ∈ ( L \ C L ) n . It is easy to see that we can find a linear combination δ of D , . . . , D r such thatnone of the coordinates of ¯ v is in ker δ . Note that δ ∈ Ann Λ(
L/C ) . Now in thedifferential field ( L ; + , · , δ ) we have ¯ v ∈ E × J ( L ) . Let K be a common differentialfield extension of L and F over C , that is, L and F can be embedded into K IFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION 11 so that the image of C under those embeddings is the same, and the appropriatediagram commutes. Such a field K exists due to the amalgamation property ofdifferential fields. Since ¯ v ∈ V ( L ) ∩ E × J ( L ) , we also have ¯ v ∈ V ( K ) ∩ E × J ( K ) . (cid:3) Generic points.
Often it is important to know that certain varieties containgeneric E J -points. In this section we state two such results. Theorem 3.7.
Let ( F ; + , · , D ) be an ℵ -saturated differentially closed field, andlet V ⊆ F n be a J -broad variety defined over a finitely generated subfield A ⊆ F .Then V ( F ) ∩ E J ( F ) contains a point generic in V over A . Theorem 3.8.
Let ( F ; + , · , D ) be an ℵ -saturated differentially closed field withfield of constants C , and let V ⊆ F n be a strongly J -broad variety defined over afinitely generated subfield C ⊆ C . Then V ( F ) ∩ E × J ( F ) contains a point genericin V over C . Furthermore, if V is also J -free then V ( F ) ∩ E × J ( F ) contains apoint generic in V over C . Theorem 3.7 follows from Theorem 1.2 and [Asl18a, Proposition 4.35], which isbased on Rabinovich’s trick to show that a Zariski open subset of an irreducible J -broad variety is isomorphic to a J -broad variety in a higher dimensional spaceand the latter contains an E J -point. The same argument can be applied to deducethe first part of Theorem 3.8 from Theorem 1.3. The second part of Theorem 3.8follows from [Asl18a, Proposition 4.29].3.5. EC for j without derivatives. In a differential field we define a binaryrelation E j ( x, y ) as the projection of E J onto the first two coordinates, in otherwords we ignore the derivatives. The theorems proved above obviously implysimilar statements for E j , in particular, Theorem 1.1. First we define j -broadvarieties. Definition 3.9. • For positive integers n ≥ k and ≤ i < . . . < i k ≤ n define a projection map π ¯ i : K n → K k by π ¯ i : (¯ x, ¯ y ) (pr ¯ i ¯ x, pr ¯ i ¯ y ) . • Let K be an algebraically closed field. An irreducible algebraic variety V ⊆ K n is j -broad if for any ≤ i < . . . < i k ≤ n we have dim π ¯ i ( V ) ≥ k . Proof of Theorem 1.1. If V ⊆ K n is j -broad then ˜ V := V × K n is obviously J -broad. Hence, by Theorem 1.2 there is a differential field extension K ⊇ F anda point (¯ z, ¯ j, ¯ j ′ , ¯ j ′′ ) ∈ ˜ V ( K ) ∩ E J ( K ) . Then (¯ z, ¯ j ) ∈ V ( K ) ∩ E j ( K ) . (cid:3) EC in fields with several commuting derivations
In this section we give a proof of EC for the exponential differential equationin fields with several commuting derivations. Strictly speaking, it is a new result since Kirby addressed the EC question only for ordinary differential fields. Notethat even in the case of a single derivation, our proof differs from Kirby’s proof(see [Kir09, Theorem 3.10]) in that we use Ax-Schanuel, rather than its proof.The proof, which is a slight generalisation of the arguments presented above, alsoworks for the j -function and establishes an EC result for E J in fields with severalcommuting derivations. Moreover, we believe that our idea of using Ax-Schanuel toprove EC is quite general and will go through in other settings too, and proving ECfor the exponential differential equation in this section supports this speculation.The EC statement for the exponential differential equation presented belowdiffers from the EC statement of [Kir09, Definition 2.30]. In fact, the latter isnot completely correct, and it was corrected in [BK18, Definition 10.3] (see also[Kir19, Fact 2.3]), which is the statement we consider in this section.In a differential field ( F ; + , · , D , . . . , D m ) we define a binary relation Exp( x, y ) by the formula V mk =1 D k y = yD k x. Fact 4.1 (Ax-Schanuel, [Ax71]) . Let ( F ; + , · , D , . . . , D m ) be a differential fieldwith field of constants C = T mk =1 ker D k . Let also ( x i , y i ) ∈ Exp( F ) , i = 1 , . . . , n, be such that x , . . . , x n are Q -linearly independent mod C , that is, they are Q -linearly independent in the quotient vector space F/C . Then td C C (¯ x, ¯ y ) ≥ n +rk( D k x i ) i,k . For a field F let G a ( F ) and G m ( F ) denote the additive and multiplicative groupsof F respectively, and for a positive integer n let G n := G n a × G n m . For a k × n matrix M of integers we define [ M ] : G n ( F ) → G k ( F ) to be the map given by [ M ] : (¯ x, ¯ y ) ( u , . . . , u k , v , . . . , v k ) where u i = n X j =1 m ij x j and v i = n Y j =1 y m ij j . Definition 4.2.
An irreducible variety V ⊆ G n ( F ) is rotund if for any ≤ k ≤ n and any k × n matrix M of integers dim[ M ]( V ) ≥ rk M . Theorem 4.3.
Let ( F ; + , · , D , . . . , D m ) be a differential field with m commutingderivations, and let V ⊆ F n be a rotund variety. Then there exists a differentialfield extension K of F such that V ( K ) ∩ Exp( K ) = ∅ . In particular, when F isdifferentially closed, V ( F ) ∩ Exp( F ) = ∅ .Proof. Let ¯ v := (¯ x, ¯ y ) be a generic point of V over F . Set K := F (¯ v ) alg . If dim V = n + l for some l ≥ then td( K/F ) = n + l .We may assume the derivations D , . . . , D m are F -linearly independent. Let ∆ := span F { D , . . . , D m } ⊆ Der(
F/C ) . Consider the space of derivations Der( K/ ∆) := { D ∈ Der(
K/C ) : D | F = λ D + . . . + λ m D m with λ i ∈ K } . IFFERENTIAL EXISTENTIAL CLOSEDNESS FOR THE j -FUNCTION 13 Extending F if necessary, we may assume that for each i there is t i ∈ F such that D i t i = 0 and D k t i = 0 whenever k = i . Therefore, for every D ∈ Der( K/ ∆) thereis a unique tuple ¯ λ ∈ K m such that D | F = λ D + . . . + λ m D m , and we denote itby ¯ λ D . The map ϕ : D ¯ λ D is a linear map ϕ : Der( K/ ∆) → K m . It is clearlysurjective and ker( ϕ ) = Der( K/F ) , hence dim Der( K/ ∆) = dim Der( K/F ) + m = n + l + m. As in the proof of Theorem 3.5, we have a sequence of inclusions
Der(
K/F ) ֒ → Der( K/ ∆) ֒ → Der(
K/C ) , and a dual sequence of surjections Ω( K/C ) ։ Ω( K/ ∆) ։ Ω( K/F ) . Consider the differential forms ω i := dy i − y i dx i ∈ Ω( K/C ) . Denote Λ( K/C ) :=span K { ω i : i = 1 , . . . , n } ⊆ Ω( K/C ) and let Λ( K/ ∆) ⊆ Ω( K/ ∆) and Λ( K/F ) ⊆ Ω( K/F ) be the images of Λ( K/C ) under the canonical surjections. Claim.
The forms ω i , i = 1 , . . . , n, are K -linearly independent in Ω( K/F ) , thatis, dim Λ( K/F ) = n . Proof.
Assume dim Λ(
K/F ) < n . Consider the annihilator Ann(Λ(
K/F )) as asubspace of Der(
K/F ) . It is clear that r := dim Ann(Λ( K/F )) = dim Ω(
K/F ) − dim Λ( K/F ) > l. The space
Ann(Λ(
K/F )) is closed under the Lie bracket, hence we can choose acommuting basis of derivations δ , . . . , δ r ∈ Ann(Λ(
K/F )) . Then δ i y k = y k δ i x k forall i, k . Let L := T ri =1 ker δ i be the field of constants. Then F ⊆ L ( K .We choose a maximal subtuple of ¯ x which is Q -linearly independent modulo L .Assume without loss of generality that ( x , . . . , x t ) is such a subtuple. Observe that t > since otherwise L = K . For every i > t there are integers β i = 0 , α i , . . . , α it and an element l i ∈ L such that α i x + . . . + α it x t + β i x i = l i . Let u i := y α i · · · y α it t · y β i i . Obviously δ k u i = u i δ k l i = 0 for all i, k , hence u i ∈ L .By rotundity of V we have td( L/F ) ≥ td( F (¯ l, ¯ u ) /F ) ≥ n − t, and by theAx-Schanuel theorem td( L (¯ x, ¯ y ) /L ) ≥ t + rk( δ i x k ) k,i = t + r. Thus td(
K/F ) = td(
K/L ) + td(
L/F ) ≥ t + r + n − t = n + r > n + l, which is a contradiction. (cid:3) Now we have dim Ann Λ( K/ ∆) = dim Ω( K/ ∆) − dim Λ( K/ ∆) = ( n + l + m ) − n = l + m and dim Ann Λ( K/F ) = dim Ω(
K/F ) − dim Λ( K/F ) = l . Choose derivations δ , . . . , δ m ∈ Ann Λ( K/ ∆) which are K -linearly independent modulo Ann Λ(
K/F ) . Each δ i | F is a K -linear combination of D k ’s, so in a matrix form δ | F = M · D, where δ and D are the column vectors of the derivations δ i and D i respectively and M ∈ GL m ( K ) (it is invertible for otherwise a K -linear combination of δ i ’s wouldbe in Der(
K/F ) ). Replacing δ with M − · δ we may assume M is the identitymatrix and hence each δ i is an extension of D i . (cid:3) The following theorem for the j -function can be proven similarly. Theorem 4.4.
Let F be a differential field with several commuting derivations,and let V ⊆ F n be a J -broad variety. Then there exists a differential field exten-sion K of F such that V ( K ) ∩ E J ( K ) = ∅ . In particular, when F is differentiallyclosed, V ( F ) ∩ E J ( F ) = ∅ . Acknowledgements.
We are grateful to the referee for useful comments thathelped us improve the presentation of the paper.
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Vahagn Aslanyan, School of Mathematics, University of East Anglia, Norwich,NR4 7TJ, UK
E-mail address : [email protected]
Sebastian Eterović, Department of Mathematics, UC Berkeley, CA 94720, USA
E-mail address : [email protected] Jonathan Kirby, School of Mathematics, University of East Anglia, Norwich,NR4 7TJ, UK
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