aa r X i v : . [ m a t h . L O ] J a n NATURAL MODELS OF THEORIES OF GREEN POINTS
JUAN DIEGO CAYCEDO AND BORIS ZILBER
Abstract.
We explicitly present expansions of the complex field which are mod-els of the theories of green points in the multiplicative group case and in the caseof an elliptic curve without complex multiplication defined over R . In fact, in bothcases we give families of structures depending on parameters and prove that theyare all models of the theories, provided certain instances of Schanuel’s conjectureor an analogous conjecture for the exponential map of the elliptic curve hold. Inthe multiplicative group case, however, the results are unconditional for genericchoices of the parameters. Introduction
The object of this paper is to explicitly give models on the complex numbersfor the theories of green points constructed in [Cay14]. The models of the theoriesof green points are expansions of the natural algebraic structure on E ( K ), where K is an algebraically closed field of characteristic zero and E is the multiplicativegroup or on an elliptic curve defined over K , by a predicate for a divisible (non-algebraic) subgroup which is generic with respect to a certain predimension function.Elements of the subgroup are called green points , by a convention introduced byPoizat. Indeed, the case where E is the multiplicative group and the subgroup isrequired to be torsion-free, corresponds to the theory of fields with green pointsconstructed by Poizat. The theories are ω -stable of Morley rank ω · Theorem.
Let ǫ = 1 + βi , with β a non-zero real number, and let Q be a non-trivialdivisible subgroup of ( R , +) of finite rank. Let G = exp( ǫ R + Q ) . Assume the Schanuel Conjecture for raising to powers in K = Q ( βi ) . Then thestructure ( C ∗ , G ) can be expanded by constants to a model of a theory of green points.In particular, ( C ∗ , G ) is ω -stable. The Schanuel Conjecture for raising to powers in a subfield K of C is an (unproven)consequence of Schanuel’s conjecture that will be discussed in Section 3. In thecases where β is generic in the o-minimal structure R exp , the Schanuel Conjecture Date : January 10, 2014. for raising to powers in K = Q ( βi ) is known to hold by a theorem of Bays, Kirbyand Wilkie ([BKW10]). The above result is therefore unconditional in those cases.In Subsection 4.6, we derive an analogous result for the theories of emerald pointsconstructed in [Cay14, Section 5]. These are variations of the theories of green pointswhere the distinguished subgroup is elementarily equivalent to the additive group ofthe integers. They are superstable, non- ω -stable, of U-rank ω · Theorem.
Let E be an elliptic curve without complex multiplication defined over C and let E = E ( C ) . Assume the corresponding lattice Λ has the form Z + τ Z and isinvariant under complex conjugation.Let ǫ = 1 + βi , with β a non-zero real number, be such that ǫ R ∩ Λ = { } . Put G = exp E ( ǫ R ) .Assume the Weak Elliptic Schanuel Conjecture for raising to powers in K := Q ( βi ) (wESC K ) holds for E . Then the structure ( E, G ) can be expanded by constantsto a model of a theory of green points. In particular, ( E, G ) is ω -stable. Let us note that the assumption that the lattice Λ has the form Z + τ Z can alwaysbe made to hold by passing to an isomorphic elliptic curve. That Λ is invariant undercomplex conjugation is a restrictive assumption. It holds, however, whenever E isdefined over R . The Weak Elliptic Schanuel Conjecture for raising to powers in K will be introduced in Section 3.The above results fit into the programme, first outlined in [Zil05a], of findingmathematically natural models for model-theoretically well-behaved theories. Theyalso provide new examples of (explicitly given) stable expansions of the complexfield. Most known examples of such structures are covered by the theorems onexpansions by small sets in [CZ01] and the green subgroups are not small.Our proofs of the main theorems follow the same strategy as in [Zil04]. Ax’stheorem on a differential version of Schanuel’s conjecture from [Ax71] plays a keyrole and some geometric arguments combine elements of complex analytic geometryand o-minimality.The paper begins with two preliminary sections: In Section 2, the axioms of thetheories of green points are recalled. Section 3 contains necessary preliminaries onstructures on the complex numbers related to exponentiation.The research presented here was part of the D.Phil. thesis of the first author,written under the supervision of the second author at the University of Oxford. Itwas funded by the Marie Curie Research Training Network MODNET.2. The theories
We shall now introduce several basic notions and state the conditions that astructure must satisfy to be a model of one of the theories of green points constructedin [Cay14].In this section, let E be the multiplicative group or an elliptic curve over a field k of characteristic 0. We use additive notation for the group operation on E .Let L E be the first-order language consisting of an n -ary predicate for each sub-variety of E n defined over k , n ≥ K extending k , we have a natural L E -structureon A := E ( K ), namely: ( E ( K ) , ( W ( K )) W ∈ L E ) . ATURAL MODELS OF THEORIES OF GREEN POINTS 3
All these structures satisfy the same complete L E -theory T E and every model of T E is of this form.Also, A is an End( E )-module. The dimension function on A given by the End( E )-linear dimension, will be denoted by lin . d . End( E ) , or simply by lin . d . . We use h Y i orspan End( E ) ( Y ) to denote the End( E )-span of a subset Y of A . Since K is algebraicallyclosed, A is divisible. The ring End( E ) is an integral domain and k E := End( E ) ⊗ Z Q is its fraction field. The quotient A/ Tor( A ) is a k E -vector space and for every Y ⊂ A , lin . d . End( E ) ( Y ) equals the k E -linear dimension of φ ( Z ) in A/ Tor( A ), where φ : A → A/ Tor( A ) is the quotient map. The pregeometry on A/ Tor( A ) given bythe k E -span induces a pregeometry on A that we shall denote by span; this meansthat for Y ⊂ A , span( Y ) = φ − (span k E ( φ ( Y ))).The theory T E is strongly minimal. The acl-dimension of a tuple b ∈ A equals thetranscendence degree over k of any normalised representation of b in homogeneouscoordinates, which we shall denote by tr . d . ( b/k ) or tr . d . k ( b ). We write tr . d . fortr . d . Q .Let L = L E ∪ { G } be the expansion of the language L E by a unary predicate G .Let C be the class of all L -structures A = ( A, G ) where A is a model of T E and G is a divisible End( E )-submodule of A .Following a convention introduced by Poizat, given an L -structure A = ( A, G ) in C , we call the elements of G green points and the elements of A \ G white points .Consider the predimension function δ defined on the finite dimensional span-closedsubsets X of any structure A ∈ C by δ ( X ) = 2 tr . d . k ( X ) − lin . d . ( X ∩ G ) . Also, for any span-closed subset Y of A , the localisation of δ at Y , δ Y , is the functiongiven by δ Y ( X ) = 2 tr . d . k ( X/Y ) − lin . d . (( X + Y ) ∩ G/Y ∩ G ) , for any finite dimensional span-closed set X . We also write δ ( X/Y ) for the value δ Y ( X ), and call it the predimension of X over Y .Associated to the predimension function δ we have the notion of strong sets. Aspan-closed subset Y of A is strong in the structure A (with respect to δ ), if forevery finite dimensional span-closed subset X of A we have δ ( X/Y ) ≥ . An arbitrary subset Y of A is said to be strong in A if span( Y ) is strong in A in theabove sense. If Y is a substructure of A and its domain Y is strong in A , then wesay Y is a strong substructure of A , and that A is a strong extension of Y .Let us fix a substructure X of a structure in C whose domain is a finite dimensionalspan-closed set. Let L X denote the expansion of the language L by constants forthe elements of X .Let C be the class of all L X -structures A X such that the L -reduct A is in theclass C and the interpretation of the constants gives a strong embedding of X into A . For A ∈ C , we identify X with the strong substructure of A consisting of theinterpretations of the constants. With this convention in place, we may omit anyexplicit mention of the constants in the notation for a structure in C , writing simply A instead of A X .Let us recall two definitions from [Cay14]. JUAN DIEGO CAYCEDO AND BORIS ZILBER
Definition 2.1.
An irreducible subvariety W of A n is said to be rotund if for every k × n -matrix M with entries in End( E ) of rank k , the dimension of the constructibleset M · W := { M · y : y ∈ W } is at least k .It is worth noting that for any subvariety W of A n and any C ⊂ A such that W is defined over k ( C ), if b is a generic point of W over k ( C ), then: W is rotund ifand only if for every k × n -matrix M with entries in End( E ) of rank k ,tr . d . ( M · b/k ( C )) ≥ k . Definition 2.2.
A structure A = ( A, G ) ∈ C is said to have the EC-property iffor every even n ≥ W of A n of dimension n , theintersection W ∩ G n is Zariski dense in W ; i.e. for every proper subvariety W ′ of W the intersection ( W \ W ′ ) ∩ G n is non-empty.It was shown in [Cay14] (see, in particular, Lemmas 3.8 and 3.17) that there is acomplete L X -theory T X whose models are precisely the structures ( A, G ) X suchthat(1) ( A, G ) X is in C (i.e. X is strong in ( A, G )), and(2) (
A, G ) has the EC-property.The theories of green points are the theories obtained in this way.3.
Some preliminaries
This section gathers several definitions and results about structures on the com-plex numbers related to exponentiation. These will be applied in the later sections.3.1.
Exponentiation and raising to powers.
Exponentiation.
Let C exp = ( C , + , · , exp) be the expansion of the complexfield by the exponential function. The Schanuel Conjecture from transcendentalnumber theory, which we state below, can be regarded as a statement about C exp . Conjecture 3.1 (The Schanuel Conjecture (SC)) . For every n and every Q -linearlyindependent tuple x ∈ C n , tr . d . ( x exp x ) ≥ n. The predimension function δ exp is defined on any tuple x ⊂ C by δ exp ( x ) := tr . d . ( x exp x ) − lin . d . Q ( x ) . The Schanuel conjecture is equivalent to the statement that for every x ⊂ C , δ exp ( x ) ≥
0. Therefore, if the SC holds, then δ exp is a proper predimension functionon C exp .In [Zil05c], a model-theoretic study of the structure C exp is carried out usingthe predimension function δ exp . Here we shall only need one aspect of that work,namely Zilber’s proof that the pregeometry associated to δ exp has the countableclosure property ([Zil05c, Lemma 5.12]). Versions of this fact will be essential in ourarguments in sections 4 and 5. We include the proof, in slightly greater detail thanin [Zil05c].Let us recall a general fact that applies to each of the predimension functions δ on C considered in this paper (for more details we refer to [Cay14, Definition 2.24and Remark 2.25]). If δ is non-negative, we have an associated dimension functiond, defined for all finite X ⊂ A byd( X ) = min { δ ( X ′ ) : X ⊂ X ′ ⊂ fin C } . ATURAL MODELS OF THEORIES OF GREEN POINTS 5 and a corresponding pregeometry cl on the set { x ∈ C : d( x ) ≤ } (with dimensionfunction d) given by: for X ⊂ fin C and x ∈ C , x ∈ cl d ( X ) ⇐⇒ d( x /X ) = 0 . Let us note the following alternative formulations of the above: x ∈ cl d ( X ) ⇐⇒ d( x /X ) = 0 ⇐⇒ d( x X ) = d( X ) ⇐⇒ there exists a tuple x ⊃ x such that δ ( x/ scl( X )) = 0 ⇐⇒ x ∈ scl( X ) or there exists a tuple x ⊃ x , cl -independent over scl( X ),such that δ ( x/ scl( X )) = 0 , where scl( X ) denotes the strong closure with respect to the predimension function δ ,i.e. the smallest strong span-closed set containing X . Later on it will be importantto know the following: first, the strong closure of any set exists; second, if a set hasfinite linear dimension then its strong closure also has finite linear dimension (andin particular is countable); the strong closure of any set is the union of the strongclosures of its finite dimensional subsets, hence the strong closure of a countable setis always countable (see [Cay14, Lemma 2.9]).Let us assume the Schanuel Conjecture for the rest of this subsubsection. Thus,the predimension function δ exp is non-negative and, we have an associated dimensionfunction d exp on C exp , and a corresponding pregeometry cl exp Definition 3.2.
For any
A ∈ C , the dimension function d associated to δ is definedfor all finite X ⊂ A by the formulad( X ) = min { δ ( X ′ ) : X ⊂ X ′ ⊂ fin A } . Remark 3.3.
The function d has the following properties: • d( ∅ ) = 0. • For all
X, Y ⊂ fin A , if X ⊂ Y then d( X ) ≤ d( Y ). • For all
X, Y, Z ⊂ fin A , if d( XY ) = d( Y ) then d( XY Z ) = d(
Y Z ). It follows that associated to d we have a closure operator with finite character cl d on A which restricts to a pregeometry on the set A := { x ∈ A : d( x ) ≤ } withdimension function d. Indeed, the operator cl d is given by: for X ⊂ fin A and x ∈ A , x ∈ cl d ( X ) ⇐⇒ d( x /X ) = 0 ⇐⇒ d( x X ) = d( X ) ⇐⇒ There exists a tuple x ⊃ x such that δ ( x/ scl( X )) = 0 ⇐⇒ x ∈ scl( X ) or there exists a tuple x ⊃ x , cl -independent over scl( X ),such that δ ( x/ scl( X )) = 0 . The following two definitions will be needed to understand cl exp in the proof ofLemma 3.7.
Definition 3.4.
Let n ≥
1. A subvariety W of C n × ( C ∗ ) n is said to be ex -rotund if for every k × n -matrix M with entries in Z of rank k , dim W ′ ≥ k , where Equivalently, for all
Y, Z ⊂ fin A and all x ∈ A , if d( xY ) = d( Y ) then d( xY Z ) = d( Y Z ). JUAN DIEGO CAYCEDO AND BORIS ZILBER W ′ is the image of W under the map from C n × ( C ∗ ) n to C k × ( C ∗ ) k given by( x, y ) ( M · x, y M ) Definition 3.5.
Let W ⊂ C n × ( C ∗ ) n be an ex-rotund variety and let B ⊂ C be suchthat W is defined over B ∪ exp( B ). Let us say that a ∈ C n is a generic realisation of W over B , if ( a, exp( a )) is a generic point of W over B ∪ exp( B ).Also, consider the following definition: Definition 3.6.
A pregeometry cl on a set A is said to have the Countable ClosureProperty (CCP) if for every finite subset X of A , the set cl( X ) is countable.Let us remark the simple fact that if a pregeometry cl has the CCP, then the localisation cl D of cl over a countable set D , i.e. the pregeometry defined by theformula cl D ( X ) = cl( D ∪ X ), also has the CCP.We can now give the statement and proof of Lemma 5.12 from [Zil05c]. Lemma 3.7.
Assume SC holds. Then the pregeometry cl exp has the CCP.Proof. Let B be a finite subset of C . We shall prove that cl exp ( B ) is countable. Bypassing to its strong closure, we may assume that B is strong with respect to δ exp .Note that for any element x ∈ C , x is in cl exp ( B ) if and only if x ∈ span Q ( B ) orthere exists x ⊃ x , Q -linearly independent over B , such that δ exp ( x/B ) = 0.It is clear that span Q ( B ) is countable, it therefore suffices to show that the set { x ⊂ C : x is Q -linearly independent over B and δ exp ( x/B ) = 0 } is also countable.Suppose x is Q -linearly independent over B and let W be the algebraic locus of( x, exp( x )) over B ∪ exp( B ). Since B is strong, the variety W is ex-rotund and,clearly, x is a generic realisation of W over B . Also, note that δ exp ( x/B ) = 0 ifand only if dim W = n . Thus, it is sufficient to prove that for every n and forevery ex-rotund variety W ⊂ C n defined over B ∪ exp( B ) of dimension n , the setof generic realisations of W over B is countable (clearly, there are only countablymany such varieties W .) This is done below.Let W ⊂ C n be an ex-rotund variety defined over B ∪ exp( B ) of dimension n .The proof of the following claim completes the proof of the lemma. Claim:
Consider the (analytic) set S = { x ∈ C n : ( x, exp x ) ∈ W } . There is an analytic set S of dimension zero contained in S such that every genericrealisation of W over B either is in S or is an isolated point of S . Indeed, the claim implies that the set of generic realisations of W over B iscountable: Since S is an analytic set of dimension zero, it consists of isolated points,it is therefore discrete and hence countable (for every discrete subset of Euclideanspace is countable). Also, the set of isolated points of S is clearly discrete and hencecountable. Proof of Claim:
Being analytic, the set S can be written as a union S ≤ i ≤ d S i where, for each i , the set S i is a complex manifold of dimension d (possibly empty)and the union S ≤ j ≤ i S j is an analytic set ([Chi89, Section 5.5]). In particular, theset S is an analytic set of dimension 0. An analytic subset of a domain U in C n is a set that locally, around every point in U , is definedas the zero set of some complex analytic functions. We call analytic subsets of C n simply analyticsets . For precise definitions see [Chi89, Section 2.1]) ATURAL MODELS OF THEORIES OF GREEN POINTS 7
Let us now show that any generic realisation of W over B in S \ S is an isolatedpoint of S .Suppose not. Then there exists a generic realisation a of W over B in S \ S thatis not an isolated point of the analytic set S .Since a is in some S i with i >
0, there exists an analytic isomorphism x : t x ( t )from an open disc D around 0 in C onto a subset of S mapping 0 to a .Set y ( t ) := exp( x ( t )). Then for every t ∈ D , ( x ( t ) , y ( t )) is in W .We can consider (the germ of) each coordinate function of x and y as an elementof the differential ring R of germs near 0 of functions which are analytic on aneighbourhood of 0. Note that the ring of constants of R is (isomorphic to) C . Using the fact that the zero set of an analytic function in one variable consistsof isolated points, it is easy to see that R is an integral domain. Thus, R embedsinto its field of fractions, F . The derivation on R extends to a derivation on F (bythe usual differentiation rule) with field of constants C ⊃ C .Since ( x, y ) ∈ W ( F ), we get thattr . d . ( x, y/B ∪ exp( B )) ≤ n. In fact, since x (0) = a is a generic realisation over B , tr . d . ( x (0) , y (0) /B ∪ exp( B )) = n , and hence tr . d . ( x, y/B ∪ exp( B )) = n. Let k ∈ { , . . . , n } be the number of independent Q -linear dependences among Dx , . . . , Dx n , i.e. k = n − lin . d . Q ( Dx , . . . , Dx n ). After a Q -linear change of coordi-nates we can assume that Dx , . . . , Dx k are all identically zero and Dx k +1 , . . . , Dx n are Q -linearly independent. Thus, x , . . . , x k are all constant, with values a , . . . , a k respectively. Since W is ex-rotund, we havetr . d . ( a , . . . , a k , exp( a ) , . . . , exp( a k ) /B ∪ exp( B )) ≥ k. Hencetr . d . ( x k +1 , . . . , x n , y k +1 , . . . , y n / C ) ≤ tr . d . ( x k +1 , . . . , x n , y k +1 , . . . , y n / C ) ≤ tr . d . ( x k +1 , . . . , x n , y k +1 , . . . , y n /B ∪ { a , . . . , a k } ∪ exp( B ∪ { a , . . . , a k } ))= tr . d . ( x k +1 , . . . , x n , y , . . . , y n /B ∪ exp( B )) − tr . d . ( x , . . . , x k , y , . . . , y k /B ∪ exp( B )) ≤ n − k. Ax’s Theorem ([Ax71, Statement (SD)]) then implies that Dx k +1 , . . . , Dx n mustbe Q -linearly dependent. This gives a contradiction. (cid:3) Remark 3.8.
The only use of the Schanuel Conjecture in Lemma 3.7 is in theassertion that cl exp is a pregeometry. By results of Kirby (Theorem 1.1 and Theorem1.2 in [Kir10]), without assuming the Schanuel conjecture, there is a countable strongsubset D of C with respect to the predimension function δ exp . Thus, unconditionally,for such D , the localisation (cl exp ) D of cl exp is a pregeometry and, by precisely thesame argument as in the proof of Lemma 3.7, has the CCP. The equivalence relation defining the germs is given by: f ∼ g , if f and g coincide on apunctured neighbourhood of 0. JUAN DIEGO CAYCEDO AND BORIS ZILBER
Raising to powers.
Let K be a subfield of C . The structure C K of raising topowers in K is the following two-sorted structure:( C , + , ( λ · ) λ ∈ K ) exp −−→ ( C , + , · ) , where the structure on the first-sort is the natural K -vector space structure, thestructure on the second sort is the usual field structure and exp is the complexexponential function.A model-theoretic study of the above structures, in analogy with the case of C exp ,has been done by Zilber in [Zil03], with additions in [Zil11] and [Zil02]. As in theprevious subsection, we are interested in a CCP result and give only a brief accountof the necessary material.Consider the predimension function δ K defined on tuples x ⊂ C by δ K ( x ) := lin . d . K ( x ) + tr . d . (exp( x )) − lin . d . Q ( x ) . Assume K has finite transcendence degree. Then, the Schanuel Conjecture impliesthat δ K ( x ) ≥ − tr . d . ( K ) for all x . Indeed, lin . d . K ( x ) ≥ tr . d . ( x/K ) ≥ tr . d . ( x ) − tr . d . ( K ), therefore: δ K ( x ) = lin . d . K ( x ) + tr . d . (exp( x )) − lin . d . Q ( x ) ≥ tr . d . ( x ) − tr . d . ( K ) + tr . d . (exp( x )) − lin . d . Q ( x ) ≥ − tr . d . ( K )where the last inequality follows from the Schanuel conjecture.Thus, SC implies the following conjecture: Conjecture 3.9 (Schanuel Conjecture for raising to powers in K (SC K )) . Let K be a subfield of C of finite transcendence degree. Then, for all x ⊂ C , δ K ( x ) ≥ − tr . d . ( K ) . The following theorem shows that a stronger version of the Schanuel Conjecturefor raising to powers in K is satisfied in the case where K is generated by powersthat are exponentially algebraically independent . This result is due to Bays, Kirbyand Wilkie; in the form below, it follows easily from their Theorem 1.3 in [BKW10]. Theorem 3.10 (Strong Schanuel Condition for K (SC ∗ K )) . Suppose K = Q ( λ ) where λ is an exponentially algebraically independent tuple of complex numbers.Then for all x ⊂ C , δ K ( x ) ≥ . For the definition of exponential algebraic independence we refer to [BKW10]; forour purposes it suffices to know the following: exponential algebraic independenceimplies algebraic independence, and, if β is a real number which is generic in theo-minimal structure R exp (i.e. which is not in dcl R exp ( ∅ )) then β is exponentiallytranscendental (i.e. the singleton { β } is exponentially algebraically independent).In particular, these two facts imply that if β ∈ R is generic in R exp , then the complexnumber βi is exponentially transcendental.Assume SC K . Then the values of the submodular predimension function δ K arebounded from below in Z . Therefore there exists a smallest strong set for δ K , namelythe strong closure of the empty set. By localising δ K over this set we obtain a non-negative predimension function. Let us denote by d K the associated dimensionfunction and by cl K the corresponding pregeometry (without explicit mention of thelocalisation). ATURAL MODELS OF THEORIES OF GREEN POINTS 9
Definition 3.11.
A subset L of C n defined by an equation of the form M · x = c, where M is a k × n -matrix with entries in K and c ∈ C n , is said to be a K -affinesubspace of C n . If C ⊂ C contains all the coordinates of c , then we say that L isdefined over C . Note that if the matrix M has rank r over K , then the dimensionof L , denoted dim L , is n − r .In analogy with Definition 2.1, in the case of green points, and Definition 3.4, inthe case of exponentiation, we have the following definition, which will be essentialin our arguments in Subsection 4.3. Definition 3.12.
A pair (
L, W ) of a K -affine subspace L of C n and a subvariety W of ( C ∗ ) n is said to be K -rotund if for any k × n -matrix m with entries in Z ofrank k we have dim m · L + dim W m ≥ k. Minor modifications of the proof of the CCP for C exp yield a proof of the CCP inthe powers case under the assumption that the SC K holds. Thus, we have: Lemma 3.13.
Assume SC K . Then the pregeometry cl K on C has the CCP. Remark 3.14.
Notice that if D is a strong subset of C with respect to δ exp con-taining K , then D is also strong with respect to δ K . This is due to the fact that forevery set D ⊂ C containing K , the inequality δ K ( x/D ) ≥ δ exp ( x/D ) holds for all x ⊂ C . Indeed, this can be seen as follows: δ K ( x/D ) = lin . d . K ( x/D ) + tr . d . (exp( x ) / exp( D )) − lin . d . Q ( x/D ) ≥ tr . d . ( x/D ) + tr . d . (exp( x ) / exp( D )) − lin . d . Q ( x/D ) ≥ tr . d . ( x exp( x ) /D exp( D )) − lin . d . Q ( x/D )= δ exp ( x/D )Thus, the result of Kirby mentioned in Remark 3.8, which provides a countablestrong set D with respect to δ exp , also gives a countable strong subset of C withrespect to δ K for any countable K , namely the strong closure of K with respect to( δ exp ) D .Also, the proof of the CCP works for proving that for any countable strong set D with respect to δ K , the localisation (cl K ) D is a pregeometry with the CCP.3.2. Exponentiation and raising to powers on an elliptic curve.
Basic setting and exponentiation.
Let E be an elliptic curve defined over asubfield k of C . Put E := E ( C ) ⊂ P ( C ). The variety E has an algebraic groupstructure with identity element [0 , ,
0] and is defined by a homogeneous equation ofthe form: zy = 4( x − e )( x − e )( x − e ) , where e , e and e are distinct complex numbers.Associated to E there is a lattice Λ = ω Z + ω Z in C and a correspondingWeierstrass function ℘ , defined for x in C \ Λ by(1) ℘ ( x ) := 1 x + X ω ∈ Λ \{ } ( 1( x − ω ) − ω ) , so that the map exp E : C → E given by z ( [ ℘ ( z ) : ℘ ′ ( z ) : 1] , if z Λ, O, if z ∈ Λ,is a group homomorphism from the additive group of C onto E . The map exp E iscalled the exponential map of E .For all x ∈ C \ Λ, ℘ satisfies the differential relation(2) ( ℘ ′ ( x )) = 4( ℘ ( x ) − e )( ℘ ( x ) − e )( ℘ ( x ) − e ) , and(3) e = ℘ ( ω , e = ℘ ( ω , e = ℘ ( ω + ω . As in Section 2, we denote by End( E ) the ring of regular endomorphisms of E andby k E its field of fractions. Also, E is an End( E )-module and we denote by lin . d . the corresponding linear dimension. Here we identify End( E ) with the subring of C consisting of all α ∈ C such that α Λ ⊂ Λ. With this convention in place, for all x ⊂ C we have lin . d . End( E ) ( x/ Λ) = lin . d . (exp E ( x )). Finally, the j -invariant of E willbe denoted by j ( E ).Let us also consider the action of complex conjugation on the above setting.Throughout, we denote by z c the complex conjugate of a complex number z . Thelattice Λ c obtained from Λ by applying complex conjugation has an associated Weier-strass function ℘ c satisfying the relation ℘ c ( z c ) = ( ℘ ( z )) c for all z Λ. Let us denoteby E c the corresponding elliptic curve. By 3, the affine part of E c is defined by theequation(4) y = 4( x − e c )( x − e c )( x − e c ) . Also, since j is the value of a rational function on e , e , e (see the proof of [Sil94,I.4.5]), j ( E c ) = j ( E ) c .3.2.2. The Elliptic Schanuel Conjecture.
The following is the Elliptic Conjecturefrom [Ber02]. There it is shown to be an instance of more general conjectures ofGrothendieck and Andr´e. We will refer to it as the
Elliptic Schanuel Conjecture(ESC) .Let us start by introducing some conventions from the theory of elliptic integrals.Given an element y ∈ E , an integral of the first kind is a preimage of y under theexponential map exp E . A period of E is an integral of the first kind of the point O ,i.e. an element of Λ. Integrals of the second kind are more difficult to describe and,although they appear in the statement of the ESC below, we will not need to usetheir definition.
Quasiperiods are integrals of the second kind of the point O . Forcomplete definitions we refer to Section I.5 in [Sil94].We assume that the generators ω and ω of the lattice Λ of periods satisfy ℑ ( ω /ω ) >
0, and let η and η be corresponding quasiperiods, so that the Legendrerelation ω η − ω η = 2 πi holds ([Sil94, I.5.2]).In the rest of this subsection, given a tuple y = ( y , . . . , y r ) of points on the curve E , let us denote by x = ( x , . . . , x r ) and z = ( z , . . . , z r ) corresponding integrals ofthe first and the second kind, respectively. Conjecture 3.15 (Elliptic Schanuel Conjecture (ESC)) . Let E , . . . , E n be pairwisenon-isogenous elliptic curves. For any tuples y ν = ( y ν , . . . , y νr ν ) of points of E ν , ATURAL MODELS OF THEORIES OF GREEN POINTS 11 ν = 1 , . . . , n , we have: (5) tr . d . ( j ( E ν ) , ω ν , ω ν , η ν , η ν , y ν , x ν , z ν ) ν ≥ X ν lin . d . k E ν ( x ν / Λ ν ) + 4 X ν (lin . d . Q k E ν ) − − n + 1In fact, we do not need to deal directly with the quasiperiods or the integrals ofthe second kind for our purposes, for we can use a consequence of the conjecturethat ignores the precise contribution of these points to the transcendence degree onthe left hand side of inequality (5) by using obvious upper bounds. Let us thereforeshow that the above conjecture implies the following simpler statement: Conjecture 3.16 (Weak Elliptic Schanuel Conjecture (wESC)) . Let E , . . . , E n be pairwise non-isogenous elliptic curves. For any tuples x ν ∈ C r ν , k E ν -linearlyindependent over Λ ν , ν = 1 , . . . , n , we have: (6) tr . d . ( j ( E ν ) , x ν , exp ν E ( x ν )) ν ≥ X ν r ν . Proof of ESC (3.15) ⇒ wESC (3.16). Let E , . . . , E n be pairwise non-isogenous el-liptic curves. For ν = 1 , . . . , n , let x ν ∈ C r ν be k E ν -linearly independent over Λ ν .Set y ν = exp ν E ( x ν ). Then, by 3.15,(7) tr . d . ( j ( E ν ) , ω ν , ω ν , η ν , η ν , y ν , x ν , z ν ) ν ≥ X ν r ν + 4 X ν (lin . d . Q k E ν ) − − n + 1 . Without loss of generality let us assume that E , . . . , E l have no CM and E l +1 , . . . , E n have CM, 0 ≤ l ≤ n . Then P ν (lin . d . Q k E ν ) − = l + ( n − l ).Thus,(8) tr . d . ( j ( E ν ) , ω ν , ω ν , η ν , η ν , y ν , x ν , z ν ) ν ≥ X ν r ν + 4 l + 2( n − l ) − n + 1 . For each ν , the Legendre relation ω ν η ν − ω ν η ν = 2 πi holds. In particular, re-stricting our attention to E , . . . , E l , this givestr . d . ( ω ν , ω ν , η ν , η ν / πi ) ν =1 ,...,l ≤ l. In the CM case, hence for ν = l +1 , . . . , n , there are further algebraic dependences.Indeed, it is clear that in this case ω ν and ω ν are Q alg -linearly dependent and, in fact,by a theorem of Masser ([Mas75][3.1 Theorem III]), 1 , ω ν , η ν , πi form a Q alg -linearbasis of the Q alg -linear span of 1 , ω ν , ω ν , η ν , η ν , πi . Thereforetr . d . ( ω ν , ω ν , η ν , η ν / πi ) ν = l +1 ,...,n ≤ n − l. Combining the last two inequalities we gettr . d . ( ω ν , ω ν , η ν , η ν ) ν =1 ,...,n ≤ l + ( n − l ) + 1 . Thus, inequality 8 implies the following:tr . d . ( j ( E ν ) , y ν , x ν , z ν ) ν ≥ (cid:0) X ν r ν + 4 l + 2( n − l ) − n + 1 (cid:1) − (cid:0) l + ( n − l ) + 1 (cid:1) . Therefore tr . d . ( j ( E ν ) , y ν , x ν ) ν ≥ X ν r ν . (cid:3) Consider the case of a single elliptic curve E defined over k ⊂ C . Let E = E ( C ).Let us define a predimension function δ exp E on C as follows: for all x ⊂ C , let δ exp E ( x ) := tr . d . ( j ( E ) , x, exp E ( x )) − lin . d . k E ( x/ Λ) . The wESC is clearly equivalent to the statement that for all x ⊂ C , δ exp E ( x ) ≥ δ exp E is non-negative. Thus, assuming the wESC, we have an associated dimension function d exp E and corresponding pregeometry cl exp E . Using the same argument as in Zilber’s proofof the CCP (3.7), this time applying the version of Ax’s theorem for the Weierstrass ℘ -functions from [Kir05], one can see that the pregeometry cl exp E has the CCP.3.2.3. Raising to powers on E . Fix a subfield K of C extending k E .The two-sorted structure E K of raising to powers in K on E is given by:( C , + , ( λ · ) λ ∈ K ) exp E −−−→ ( E, ( W ( C )) W ∈ L E ) . where the first sort has the natural K -vector field structure, the second sort has thealgebraic structure on E , and the map exp E is the exponential map of E .Consider the predimension function δ E,K defined on tuples x ⊂ C by δ E,K ( x ) = lin . d . K ( x ) + tr . d . ( j ( E ) , ℘ ( x )) − lin . d . k E ( x/ Λ) . If K has finite transcendence degree, then the Weak Elliptic Schanuel Conjecture(3.16) implies that the inequality δ E,K ( x ) ≥ − tr . d . ( K ) holds for all x ⊂ C .Let us state this consequence of the wESC for a single elliptic curve E as anindependent conjecture. Conjecture 3.17 (Weak ESC for raising to powers in K on E (wESC K )) . Let E be an elliptic curve. Let K is a subfield of C extending k E of finite transcendencedegree. Then for all x ⊂ C , δ E,K ( x ) ≥ − tr . d . ( K ) . Assume wESC K holds. Then, by localising over the strong closure of the emptyset, we obtain a non-negative predimension function from δ E,K , for which we havean associated dimension function, which we shall denote d
E,K , and pregeometry,which will be denoted by cl
E,K . The same argument as in the proof of 3.7, usingthe version of Ax’s theorem for Weierstrass ℘ -functions from [Kir05], shows that forany countable K , cl E,K has the CCP.Let us extend Definition 3.12 from the multiplicative case to include the ellipticcurve case. Since there is no space for confusion, we keep the same terminology.
Definition 3.18.
A pair (
L, W ) of a K -affine subspace L of C n and an algebraicsubvariety W of E n is said to be K -rotund if for any k × n -matrix m with entries inEnd( E ) of rank k we have we havedim m · L + dim m · W ≥ k. Models on the complex numbers: the multiplicative group case
In this section, we will find models for the theories of green points in the multi-plicative group case.
ATURAL MODELS OF THEORIES OF GREEN POINTS 13
The Models.
Throughout this section, let E = G m and A = E ( C ) = C ∗ .Since we work in the multiplicative group, we shall use multiplicative notation. Wealso use the expressions multiplicatively (in)dependent instead of End( E )-linearly(in)dependent.Let ǫ ∈ C \ ( R ∪ i R ) and let Q be a non-trivial divisible subgroup of ( R , +) offinite rank. Put G = exp( ǫ R + Q ) . Note that G is a divisible subgroup of C ∗ .We assume henceforth that ǫ is of the form 1 + βi with β a non-zero real number,for we can always replace any ǫ ∈ C \ ( R ∪ i R ) for one of this form giving rise to thesame G .Consider the L -structure ( C ∗ , G ). The following theorem is the main result ofthis section. Theorem 4.1.
Let ǫ = 1 + βi , with β a non-zero real number, and let Q be anon-trivial divisible subgroup of ( R , +) of finite rank. Let G = exp( ǫ R + Q ) . Assume SC K for K = Q ( βi ) . Then:(1) For every tuple c ⊂ C ∗ , there exists a tuple c ′ ⊂ C ∗ extending c , such that c ′ is strong with respect to the predimension function ( δ G ) c ′ . If c ⊂ G , then wecan find such a c ′ also contained in G .(2) The structure ( C ∗ , G ) has the EC-property. Therefore, for every tuple c ⊂ G ,strong with respect to ( δ G ) c , the structure ( C ∗ , G ) X is a model of the theory T X , where X = span( c ) with the structure induced from ( C ∗ , G ) . The above theorem follows immediately from Propositions 4.3 and 4.4 below.Subsections 4.2 and 4.3 are devoted to the corresponding proofs.4.2.
The Predimension Inequality.
In this subsection we prove the first part ofTheorem 4.1. The proof here improves upon the corresponding one in [Zil04].
Lemma 4.2.
Let K = Q ( βi ) and assume SC K .Then for all y ∈ ( C ∗ ) n , we have δ G ( y ) ≥ − . d . Q Q − tr . d . ( K ) .Proof. We may assume y ∈ G n and is multiplicatively independent. Let x ∈ C n besuch that exp( x ) = y with x = ǫt + q , t ∈ R n , q ∈ Q n . Note that x is Q -linearlyindependent over the kernel of exp.Since complex conjugation is a field automorphism of C , we have(9) 2 tr . d . ( y ) = tr . d . ( y ) + tr . d . ( y c ) . Also,(10) tr . d . ( y ) + tr . d . ( y c ) ≥ tr . d . ( yy c ) = tr . d . (exp( x ) exp( x c )) . By the SC K , lin . d . K ( xx c ) + tr . d . ( yy c ) − lin . d . Q ( xx c ) ≥ − tr . d . ( K )and therefore(11) tr . d . ( yy c ) ≥ lin . d . Q ( xx c ) − lin . d . K ( xx c ) − tr . d . ( K )Combining 9,10 and 11, we obtain2 tr . d . ( y ) ≥ lin . d . Q ( xx c ) − lin . d . K ( xx c ) − tr . d . ( K ) . Thus, in order to prove the lemma, it is sufficient to show that the differencelin . d . Q ( xx c ) − lin . d . K ( xx c ) is always at least n − . d . Q Q .Let l := lin . d . Q Q . Since x = ǫt + q , we have x c = ǫ c t + q . Also: ǫ c ǫ = 1 − βi βi ∈ Q ( βi ) = K. From this we obtain the following upper bound for lin . d . K ( xx c ):(12) lin . d . K ( xx c ) ≤ lin . d . K ( ǫt, q ) ≤ n + lin . d . Q Q = n + l. We now need to bound lin . d . Q ( xx c ) from below. Note that the values lin . d . Q ( xx c )and lin . d . K ( xx c ) do not change if we replace x by any x ′ with the same Q -linearspan (and x c by x ′ c accordingly). It follows that we can assume that for every i ∈ { l + 1 , . . . , n } , q i = 0. Indeed, one can apply appropriate regular Q -lineartransformations to x (and accordingly to x c ) to reduce to this case.Since x is linearly independent, in particular we have lin . d . Q ( x l +1 , . . . , x n ) = n − l ,i.e. lin . d . Q ( ǫt l +1 , . . . , ǫt n ) = n − l . Moreover, since ǫ R ∪ i R , ǫ and ǫ c are R -linearlyindependent. Thereforelin . d . Q ( ǫt l +1 , . . . , ǫt n , ǫ c t l +1 , . . . , ǫ c t n ) = 2( n − l ) . Thus,(13) lin . d . Q ( xx c ) ≥ lin . d . Q ( x l +1 , . . . , x n , x cl +1 , . . . , x cn ) ≥ n − l. From 12 and 13 we concludelin . d . Q ( xx c ) − tr . d . ( xx c ) ≥ (2 n − l ) − ( n + l ) = n − l. (cid:3) Proposition 4.3.
Assume SC K for K = Q ( βi ) . Then for every tuple c ⊂ C ∗ ,there exists a tuple c ′ ⊂ C ∗ , extending c , such that c ′ is strong with respect to thepredimension function ( δ G ) c ′ . If c ⊂ G , then we can find such a c ′ also contained in G .Proof. By Lemma 4.2, the set of values of δ G on ( C ∗ , G ) is bounded from below in Z . We can therefore find a tuple c such that δ G ( c ) is minimal. Since for everyspan-closed set X , δ G ( X ) ≥ δ G ( X ∩ G ), we can find such c with all its coordinatesin G . Clearly, c is strong for δ G , hence the localisation ( δ G ) c is a non-negativepredimension function on C ∗ .For every c ⊂ C ∗ , let c ′ ⊂ C ∗ be a tuple containing both c and c that generatesthe strong closure of c with respect to ( δ G ) c . The tuple c ′ is, by definition, strongfor ( δ G ) c . Since c ⊃ c ′ , it follows that c ′ is strong for ( δ G ) c ′ . It is easy to see, thatif c is contained in G , then c ′ can be taken to be contained in G . (cid:3) Existential Closedness.
This subsection is devoted to the proof of the fol-lowing proposition:
Proposition 4.4.
The structure ( C ∗ , G ) has the EC-property. Therefore, for everytuple c ⊂ G , strong with respect to ( δ G ) c , the structure ( C ∗ , G ) X is a model of thetheory T X , where X = span( c ) with the structure induced from ( C ∗ , G ) . For the rest of Subsection 4.3, let us fix an even number n ≥ V ⊂ ( C ∗ ) n of dimension n defined over k ( C ) for some finite C ⊂ C . Weneed to show that the intersection V ∩ G n is Zariski dense in V . ATURAL MODELS OF THEORIES OF GREEN POINTS 15
Let us define the set X = { ( s, t ) ∈ R n : exp( ǫt + s ) ∈ V } . Note that if ( s, t ) is in
X ∩ ( Q n × R n ), then the corresponding point y := exp( ǫt + s )is in V ∩ G n . Thus, in order to find points in the intersection V ∩ G n , we shall lookfor points ( s, t ) in X with s ∈ Q n .Our strategy for this is to find an implicit function for X defined on an open set S ⊂ R n , assigning to every s ∈ S a point t ( s ) ∈ R n such that ( s, t ( s )) ∈ X . Since Q n is dense in R n , the intersection S ∩ Q n is non-empty, and therefore we can findpoints ( s, t ( s )) in X with s ∈ Q n .Let R be the expansion of the real ordered field by the restrictions of the realexponential function and the sine function to all bounded intervals with rationalend-points, and by constants for the real and imaginary parts of the elements of Q ( C ).Since R is an expansion by constants of a reduct of R an , R is o-minimal. Notethat the set X is locally definable in R , i.e. its intersection with any bounded boxwith rational endpoints is a definable set in R .Let us briefly introduce some conventions and basic facts from dimension theory ino-minimal structures. Firstly, if R is o-minimal, the definable closure dcl R coincideswith the algebraic closure acl R and it is a pregeometry (that dcl R satisfies theexchange axiom follows from the Monotonicity Theorem ([vdD98, 3.1.2])). Thedimension function associated to the pregeometry dcl R will be denoted by dim R .For expansions of the reals, we have the following key fact: Suppose R is anexpansion of the real ordered field in a countable language. Then for any X ⊂ R n that is locally definable in R over a countable set A we havemax x ∈ X dim R ( x/A ) = dim R X, where dim R X is the topological dimension of X , i.e. the maximum k ≤ n suchthat for some coordinate projection π from R n to R k , the set π ( X ) has interior. If X is a real analytic set, then dim R X is also its real analytic dimension , i.e. themaximum k such that for some x ∈ X and open neighbourhood V x of x , X ∩ V x isa real analytic submanifold of R n of dimension k . (The first part of the above factfollows easily from the Baire Category Theorem, as noted in [HP94, Lemma 2.17].The second part is a standard fact in real analytic geometry.)If X ⊂ R n is a locally definable set in R over a A ⊂ R , an element b of X is saidto be generic in X over A ifdim R ( b/A ) = max x ∈ X dim R ( x/A ) . Our proof of Proposition 4.4 relies on the following lemma, whose proof we post-pone until the next subsection.
Lemma 4.5 ( Main Lemma ) . Suppose ( s , t ) is an R -generic point of X , i.e. dim R ( s , t ) = dim R X = n . Then dim R ( s ) = n . Let us now continue with the proof of the existential closedness, using the MainLemma.
Lemma 4.6.
Suppose ( s , t ) is an R -generic point of X . There is a continuous R -definable function s t ( s ) defined on a neighbourhood S ⊂ R n of s and takingvalues in R n such that for all s ∈ S , the point y ( s ) := exp( ǫt ( s ) + s ) is in V . Proof.
Let π : R n → R n be the projection onto the first n coordinates.Let X be the intersection of X and a box with rational end-points containing( s , t ). By the Main Lemma (4.5), dim R ( s ) = n . Since π ( X ) is definable in R , wehave dim R π ( X ) = max s ∈ π ( X ) dim R ( s ) ≥ dim R ( s ) = n . Therefore dim R π ( X ) = n , and hence the set π ( X ) contains an open neighbourhood S of s .By the definable choice property of o-minimal expansions of ordered abeliangroups ([vdD98, 6.1.2]), there is an R -definable map t : π ( X ) → R n such thatfor all s ∈ π ( X ), ( s, t ( s )) is in X . In particular, for all s ∈ S , ( s, t ( s )) ∈ X , i.e. y ( s ) := exp( ǫt ( s ) + s ) is in V .The o-minimality of R also gives that the set of points where the R -definablefunction t is discontinuous is R -definable and of dimension strictly lower than n (this follows from the C -Cell Decomposition Theorem [vdD98, 7.3.2], together withthe fact that the boundary of any subset of R n has dimension strictly less than n [vdD98, 4.1.10]). Thus, by making S smaller if necessary, we may assume t iscontinuous on S . (cid:3) Proof of Proposition 4.4.
Let V ′ be a proper subvariety of V . We need to see thatthe intersection ( V \ V ′ ) ∩ G n is non-empty.Extending C if necessary, we may assume that V and V ′ are defined over C .Take an element y of V \ V ′ with dim R ( y ) = dim R V = n . Let x ∈ C n be suchthat exp( x ) = y and let t , s ∈ R n be such that x = ǫt + s .Note that dim R ( s , t ) = dim R ( x ) = dim R ( y ) = n . Hence ( s , t ) is R -genericin X .Let S and the map s t ( s ) be as provided by Lemma 4.6 for R and ( s , t ).Consider the map s y ( s ) := exp( ǫt ( s ) + s ) defined on S . This map is continuous,hence y − ( V ′ ) is a closed subset of S not containing s . Thus, S ′ = S \ y − ( V ′ ) isan open neighbourhood of s .Since Q n is dense in R n , we can take a point q in S ′ ∩ Q n , and thus obtain acorresponding point y ( q ) in ( V \ V ′ ) ∩ G n . (cid:3) Proof of the Main Lemma.
The image of V under complex conjugation, V c ,plays an important role in our proof of the Main Lemma. Since complex conjugationis a field isomorphism, V c is also an irreducible algebraic variety defined over theset C c . Extending C if necessary, we may assume that V c is also defined over C . Notation 4.7.
For a tuple x of variables or complex numbers, the expression ¯ x will denote another tuple of variables or complex numbers, respectively, of the samelength, bearing no formal relation to the former. This notation is meant to implythat we are particularly interested in the case where x is a complex number and ¯ x equals x c .Throughout this subsection, let K = Q ( βi ). Definition 4.8.
For s ∈ C n , we define the set L s = { ( x, ¯ x ) ∈ C n : ( x + ¯ x ) + β − i ( x − ¯ x ) = 2 s } . Remark 4.9.
Note that β − i = − ( βi ) − ∈ K , hence L s is a K -affine subspace. Remark 4.10.
Suppose s is in R n . Then, for all x ∈ C n , the point ( x, x c ) belongsto L s if and only if x = ǫt + s for some t ∈ R n . ATURAL MODELS OF THEORIES OF GREEN POINTS 17
To see this, let x ∈ C n be given, and let t ∈ C n be such that x = ǫt + s . Then:( x, x c ) ∈ L s ⇐⇒ ( x + x c ) + β − i ( x − x c ) = 2 s ⇐⇒ x ) + β − i (2 i Im( x )) = 2 s ⇐⇒ (Re( ǫt ) + s ) − β − Im( ǫt ) = s ⇐⇒ Re( ǫt ) − β − Im( ǫt ) = 0 ⇐⇒ (Re( t ) − β Im( t )) − β − (Im( t ) + β Re( t )) = 0 ⇐⇒ ( β + β − ) Im( t ) = 0 ⇐⇒ Im( t ) = 0 ⇐⇒ t ∈ R n . Lemma 4.11.
Suppose s ∈ R n . Then for all linearly independent ( m , n ) , . . . , ( m k , n k ) ∈ Z n ( m i , n i ∈ Z n ), we have dim( m, n ) · L s ≥ k . Proof.
Suppose ( m , n ) , . . . , ( m k , n k ) ∈ Z n are linearly independent ( m i , n i ∈ Z n ).Let D be any countable set over which L s is defined and let t ∈ R n be such thatlin . d . K ( t/D ) = n . For x = ǫt + s and ¯ x = x c = ǫ c t + s , the tuple ( x, ¯ x ) is in L s , as4.10 shows. Then, we have:dim( m, n ) · L s ≥ lin . d . K (( m, n ) · ( x, ¯ x ) /D )= lin . d . K (( m , n ) · ( x, ¯ x ) , . . . , ( m k , n k ) · ( x, ¯ x ) /D )= lin . d . K (( m ′ , n ′ ) · ( t, βit ) , . . . , ( m ′ k , n ′ k ) · ( t, βit ) /D )where m ′ i = m i + n i and n ′ i = m i − n i , for all i = 1 , ..., k . Since m i = ( m ′ i + n ′ i )and n i = ( m ′ i − n ′ i ), the matrix ( m ′ , n ′ ) has the same rank as ( m, n ), that is k .Therefore we can take a matrix M ∈ GL k ( Z ) and t ′ = ( t j , . . . , t j l , βit j l +1 , . . . , βit j k ),with 1 ≤ l ≤ k , such thatlin . d . K (( m ′ , n ′ ) · ( t, βit ) , . . . , ( m ′ k , n ′ k ) · ( t, βit ) /D ) = lin . d . K ( M · t ′ /D ) . Thus, dim( m, n ) · L s ≥ lin . d . K ( M · t ′ /D ) . But note that lin . d . K ( M · t ′ /D ) is at least k , for we havelin . d . K ( M · t ′ /D ) = lin . d . K ( t ′ /D ) ≥ max { lin . d . K ( t j , . . . , t j l /D ) , lin . d . K ( βit j l +1 , . . . , βit j k /D ) } = max { l, k − l } ≥ k . Therefore, dim( m, n ) · L s ≥ k (cid:3) Lemma 4.12.
Let s ∈ R n . Then the pair ( L s , V × V c ) is K -rotund.Proof. Suppose ( m , n ) , . . . , ( m k , n k ) ∈ Z n are linearly independent ( m i , n i ∈ Z n ).The rotundity of V implies that the variety V × V c is also rotund. Hence dim( V × V c ) ( m,n ) ≥ k .Also, by Lemma 4.11, dim( m, n ) · L s ≥ k .Therefore, we have:dim( m, n ) · L s + dim( V × V c ) ( m,n ) ≥ k k k. Thus, the pair ( L s , V × V c ) is K -rotund. (cid:3) Proof of the Main Lemma.
Consider the set L s ∩ log( V × V c ) . It is an analytic subset of C n containing the point ( x , ( x ) c ). Since every analyticset can be written as the union of its irreducible components and this union is locallyfinite ([Chi89, Section 5.4]), there exist a neighbourhood B of ( x , ( x ) c ), a positiveinteger l and irreducible analytic subsets S , . . . , S l of B containing ( x , ( x ) c ) suchthat L s ∩ log( V × V c ) ∩ B = S ∪ · · · ∪ S l . We may assume B is a box with rational end-points. Claim.
Every S i has complex analytic dimension . Before proving the claim, let us show how the lemma follows. The claim impliesthat each S i is a closed discrete subset of B ; since B is bounded, each S i must thenbe finite. Being the union of the S i , the set L s ∩ log( V × V c ) ∩ B is therefore finite,and it is clearly R -definable over s . Thus, the singleton { ( x , ( x ) c ) } is R -definableover s as the intersection of L s ∩ log( V × V c ) ∩ B and a sufficiently small R -definableopen box around ( x , ( x ) c ). Therefore dim R ( s ) = dim R ( x ) = n . Proof of the claim.
Suppose towards a contradiction that there exists i such thatthe set S := S i is of positive dimension.Let us show that there are uncountably many points in S whose image underexponentiation is a generic point of V × V c over C .To see this, suppose V ′ is a proper subvariety of V × V c over C . Note that( y , ( y ) c ) is a generic point of V × V c over C , for we havetr . d . ( y , ( y ) c /C ) = tr . d . (Re( y ) , Im( y ) /C ) ≥ dim R (Re( y ) , Im( y )) = n = dim V × V c . Hence ( x , ( x ) c ) does not belong to log V ′ , and therefore S ∩ log V ′ is an analyticsubset of B properly contained in S . Then, by the irreducibility of S , for any such V ′ , S ∩ log V ′ is nowhere-dense in S . Since S has positive dimension we can apply theBaire Category Theorem to conclude that there exist uncountably many ( x, ¯ x ) in S that do not belong to log V ′ for any such V ′ , i.e. their images under exponentiationare generic points of V × V c over C .Let D be a countable strong subset of C with respect to δ K (provided by Re-mark 3.14). Let D ′ be the strong closure of log C ∪ s with respect to ( δ K ) D .For any tuple z ⊂ C ∗ , if δ K ( z/D ′ ) ≤ z lie in cl K ( D ′ ).But cl K ( D ′ ) is countable, because D ′ is countable and (cl K ) D has the CountableClosure Property, so there can be no more than countably many tuples z with δ K ( z/D ′ ) ≤
0. Thus, we can find ( x, ¯ x ) ∈ L s such that (exp( x ) , exp(¯ x )) is a genericpoint of V × V c over C and δ K ( x, ¯ x/D ′ ) > < δ K ( x ¯ x/D ′ ) ≤ dim L s ∩ N + dim( V × V c ) ∩ exp N − dim N, where N is the minimal Q -affine subspace over D ′ containing the point ( x, ¯ x ).Since dim L s = dim( V × V c ) = n , it immediately follows from the inequalityabove that N cannot be the whole of C n , as in that case the right hand side wouldbe 0. Therefore dim N < n . ATURAL MODELS OF THEORIES OF GREEN POINTS 19
Thus, there exist k ≥ m , . . . , m k ∈ Z n such that N isa translate of the subspace of C n defined by the system of equations m i · ( z, ¯ z ) = 0( i = 1 , . . . , k ). Note that dim N = 2 n − k .Note that ( V × V c ) ∩ exp N is a generic fibre of the map ( ) m on ( V × V c ), forit contains the generic point ( y, ¯ y ) of V × V c over C . The addition formula for thedimension of fibres of algebraic varieties then givesdim( V × V c ) m = dim V × V c − dim( V × V c ) ∩ exp N. Also, by the addition formula for the dimension of K -affine subspaces,dim m · L s = dim L s − dim L s ∩ N. Adding up the two equations,dim m · L s + dim( V × V c ) m = 2 n − (dim L s ∩ N + dim( V × V c ) ∩ exp N )Using (14) we getdim m · L s + dim( V × V c ) m < n − dim N = k. This implies that the pair ( L s , V × V c ) is not K -rotund, contradicting 4.12. (cid:3)(cid:3) The question of ω -saturation. A natural question which we are unable toanswer is whether the model ( C ∗ , G ) is ω -saturated. Here we present two remarkson the issue.First we show that, assuming the unproven CIT with parameters (Conjecture 4.13below), we can prove an a priori stronger version of Proposition 4.4 which is impliedby ω -saturation. In fact, if the CIT with parameters holds, then a stronger form ofthe EC-property holds in all models of T .The following is the statement of the CIT with parameters, it is a consequenceof the Conjecture on Intersections with Tori (CIT) (Conjecture 1 in [Zil02]). Fordetails, see Theorem 1 in [Zil02].
Conjecture 4.13 (CIT with parameters) . For every k ≥ , every subvariety W ( x, y ) of ( C ∗ ) n + k defined over Q and every c ∈ ( C ∗ ) k , there exists a finite collection H , . . . , H s of cosets of proper algebraic subgroups of ( C ∗ ) n with the following prop-erty:for every coset H of a proper algebraic subgroup of ( C ∗ ) n , if S is an atypical compo-nent of the intersection of W ( x, c ) and H (i.e. dim S > dim W ( x, c ) + dim H − n ),then for some i ∈ { , . . . , s } , S is contained in H i . Proposition 4.14.
Assume the CIT with parameters. Then every model of T hasthe following strong EC-property : for any rotund variety V ⊂ ( K ∗ ) n of dimension n defined over a finite set C , there exists a generic of V over C in G n .Proof. Let ( K ∗ , G ) be a model of T . Let V be a rotund variety of dimension n defined over a finite set C .It is sufficient to find a proper subvariety V ′ of V such that for any y ∈ V ∩ G n ,if y does not lie in V ′ then y is a generic point of V over C . Indeed, that ( K ∗ , G )satisfies the EC-property guarantees that we can find a point y ∈ ( V \ V ′ ) ∩ G n , and y would then be a generic point of V over C .Without loss of generality we assume that C is strong in A . Then for any y ∈ G n δ G ( y/C ) ≥
0. In particular, for any y in V ∩ G n , if y is not a generic point of V over C then y has to be multiplicatively dependent over C . Thus, it is enough to find a proper subvariety V ′ of V over C such that for every y ∈ G n ∩ V , if y ismultiplicatively dependent over C then y is in V ′ .By the CIT with parameters, there exist cosets H , . . . , H s of proper algebraicsubgroups of ( K ∗ ) n such that any atypical irreducible component of the intersectionof V and a coset of a proper algebraic subgroup of ( K ∗ ) n is contained in some H i .Let V ′ = V ∩ S i H i . We shall now show that V ′ has the required property.Suppose y ∈ G n ∩ V is multiplicatively dependent over C , and let us see that y thenbelongs to V ′ . Let H be the smallest coset of a proper algebraic subgroup that isdefined over C and contains y . Let c H denote the codimension of H , then c H ≥ Y be an irreducible component of V ∩ H containing y . Since V ∩ H is definedover C , Y is defined over Q ( C ) alg . Then, by the predimension inequality over C ,2 dim Y − dim H ≥
0. Therefore we have2 dim Y ≥ dim H = n − c H = 2 dim V − c H , and consequently, dim Y ≥ dim V − c H > dim V − c H . Hence dim
Y > dim V − c H , which means that Y is an atypical irreducible componentof the intersection V ∩ H .Indeed, the CIT with parameters then tells us that y must belong to one of the H i , and thus to V ′ . (cid:3) In the following proposition, we freely use notions and facts from [Cay14].
Proposition 4.15.
Assume the CIT with parameters. Suppose ( K ∗ , G ) is a modelof T where G has infinite dimension for the dimension function associated to δ .Then ( K ∗ , G ) is ω -saturated.Proof. In the light of [Cay14, Proposition 3.31], it is sufficient to show that ( K ∗ , G )is rich .By the previous lemma, the CIT assumption implies that ( K ∗ , G ) satisfies the thestrong EC-property, which means that the richness property holds for prealgebraicminimal extensions .The assumption on the dimension of G implies that that the richness propertyalso holds for green generic minimal extensions . This amounts to proving that forany finite strong subset C of K ∗ , there exists b ∈ K ∗ with d G ( b/C ) = 1. But this isclear since d G ( G ) is infinite and C is finite.For minimal white generic extensions we need to find, for any C as before, anelement b ∈ K ∗ with d G ( b/C ) = 2. We proceed by taking b and b with d G ( b /C ) =d G ( b /Cb ) = 1 and setting b = b + b .It is sufficient to show that δ G ( b , b /C, b + b ) = 0. Indeed, we then get0 ≤ d G ( b , b /C, b + b ) ≤ δ G ( b , b /C, b + b ) = 0 , so d G ( b , b /C, b + b ) = 0, and hence d G ( b + b /C ) = d G ( b , b /C ) = 2.Now the calculation of δ G ( b , b /C, b + b ): By definition, δ G ( b , b /C, b + b ) = 2 tr . d . ( b , b /C, b + b ) − mult . d . ( b , b /C, b + b ) . It is easy to see that tr . d . ( b , b /C, b + b ) = 1 . Also, mult . d . ( b , b /C, b + b ) = 2 , ATURAL MODELS OF THEORIES OF GREEN POINTS 21 because the variety defined by the equation X + Y = b + b is rotund. Thus, δ G ( b , b /C, b + b ) = 2(1) − (cid:3) Unfortunately, it is not clear that the dimension d G ( G ) is infinite in our model( C ∗ , G ). Note that this would immediately follow if one could show that the corre-sponding pregeometry on the uncountable set G has the CCP.4.6. Emerald points.
In [Zil05b], a connection is established between the construc-tion of noncommutative tori, which are basic examples of non-commutative spaces,and the model theory of the expansions of the complex field by a multiplicativesubgroup of the form H = exp( ǫ R + q Z ) , where ǫ = 1 + iβ and β and q are non-zero real numbers such that βq and π are Q -linearly independent.In order to prove that such structures are superstable, in [Cay14, Section 5], avariant of the theories of green points was considered in which the distinguishedsubgroup is not divisible, but elementarily equivalent to the additive group of theintegers instead. The modified theories were named theories of emerald points andshown to be superstable. It remained to show that the above structures are in factmodels of the constructed theories.In this subsection we remark that the arguments in this section for the green casealso yield the fact that the above structures are indeed models of the theories ofemerald points (after adding constants for the elements of a strong set), providedthe Schanuel Conjecture for raising to powers in K = Q ( βi ) holds. Theorem 4.16.
Let β and q non-zero real numbers such that βq and π are Q -linearly independent. Let ǫ = 1 + βi and H = exp( ǫ R + q Z ) . Assume SC K holds for K = Q ( βi ) . Then:(1) For every tuple c ⊂ C ∗ , there exists a tuple c ′ ⊂ C ∗ extending c , such that c ′ is strong with respect to the predimension function ( δ H ) c ′ .(2) The structure ( C ∗ , H ) has the EC-property. Therefore, for every tuple c ⊂ C ∗ , strong with respect to ( δ H ) c , the structure ( C ∗ , H ) X is a model of thetheory T = T X from [Cay14, Section 5] , where X = span( c ) with thestructure induced from ( C ∗ , H ) . The first part of the theorem follows directly from the analogous statement inthe green case, by Remark 5.2 of [Cay14]. For the second part of the theorem, theproof of the analogous statement in the green case applies, simply using the densityof q Z + πβ Z in R , instead of that of the subgroup Q , at the very end of the proof.5. Models on the complex numbers: the elliptic curve case
In this section we find models for the theories of green points in the case ofan elliptic curve without complex multiplication and whose lattice of periods isinvariant under complex conjugation, under the assumption that the Weak SchanuelConjecture for raising to powers on the elliptic curve holds.
The Models.
Let us fix an elliptic curve E without complex multiplication.Let E = E ( C ). We use the conventions introduced in Subsection 3.2.Let ǫ ∈ C ∗ be such that ǫ R ∩ Λ = { } . Put G = exp E ( ǫ R ). Remark 5.1.
Note that G is a divisible subgroup of E . Since E has no CM, k E = Q and, for any y ⊂ E , span( y ) is the divisible hull of the subgroup generated by y .Also, G is dense in E in the Euclidean topology. To see this notice the following: G = exp E ( ǫ R ) = exp E ( ǫ R + Λ) = exp E (Γ + Z + α Z ) , for α = Re( τ ) − Re( ǫ )Im( ǫ ) Im( τ ) ∈ R . Since Γ ∩ Λ = { } , α is irrational. It follows that Z + α Z is dense in R . Therefore the set G = exp E ( ǫ + Z + α Z ) is dense in E in theEuclidean topology.We now state the main theorem of this section. Theorem 5.2.
Let E be an elliptic curve without complex multiplication and let E = E ( C ) . Assume the corresponding lattice Λ has the form Z + τ Z and Λ = Λ c .Let ǫ = 1 + βi , with β a non-zero real, be such that ǫ R ∩ Λ = { } . Put G =exp E ( ǫ R ) .Let K = Q ( βi ) and assume the Weak Elliptic Schanuel Conjecture for raising topowers in K (wESC K ) holds for E .Then:(1) For every tuple c ⊂ E , there exists a tuple c ′ ⊂ E extending c , such that c ′ is strong with respect to the predimension function ( δ G ) c ′ . If c ⊂ G , then wecan find such a c ′ also contained in G .(2) The structure ( E, G ) has the EC-property. Therefore, for every tuple c ⊂ G ,strong with respect to ( δ G ) c , the structure ( E, G ) X is a model of the theory T , where X = span( c ) with the structure induced from ( E, G ) . Let us make some remarks about the hypotheses of the theorem. Firstly, assumingthat the lattice Λ has generators ω = 1 and ω = τ is not truly restrictive, for thiscan always be achieved by passing to an isomorphic elliptic curve. Secondly, theassumptions of E having no CM and Λ being invariant under complex conjugation arereal restrictions on the generality of the result. The first assumption is essential, sincewe do not have an appropriate End( E )-submodule of E that serves as analogue of thesubgroup G defined above in the CM case. The second is necessary in our argumentsfor proving both the predimension inequality and the existential closedness for thestructure ( E, G ). Let us remark that the two conditions hold for any non-CM ellipticcurve defined over R . Also note that, by the remarks at the end of Subsubsection3.2.1, the assumption that Λ = Λ c implies that E = E c and j ( E ) c = j ( E ). Finally,let us recall that our assumption that the wESC K (3.17) holds for the single ellipticcurve E means the following: for any tuple x of complex numbers,lin . d . K ( x ) + tr . d . ( j ( E ) , ℘ ( x )) − lin . d . Q ( x/ Λ) ≥ − tr . d . ( K ) . For the rest of this section, we work under the hypotheses of the theorem, thatis: E has no CM, Λ = Z + τ Z , and Λ = Λ c . Also, ǫ = 1 + βi with β a non-zero realand ǫ R ∩ Λ = { } , and G = exp E ( ǫ R ). We set K = Q ( βi ) and assume the WeakElliptic Schanuel Conjecture for raising to powers in K holds for E .As in the previous section, we divide the proof of the theorem into the proofs oftwo propositions, Propositions 5.4 and 5.5. ATURAL MODELS OF THEORIES OF GREEN POINTS 23
The Predimension Inequality.
In this subsection we prove the first part ofTheorem 5.2.
Lemma 5.3.
For any tuple y ⊂ E , δ G ( y ) ≥ − − tr . d . ( K ) − . d . ( k ) .Proof. It is sufficient to show that for any n and any y ∈ G n with lin . d . Q ( y ) = n ,we have 2 tr . d . k ( y ) ≥ n − − tr . d . ( K ) − . d . ( k ) . Fix such n and y . Let x ∈ ( ǫ R ) n be such that exp E ( x ) = y . Notice that x is Q -linearly independent over Λ.Note the following2 tr . d . ( j ( E ) , ℘ ( x )) ≥ tr . d . ( j ( E ) , ℘ ( x ) , ( ℘ ( x )) c )= tr . d . ( j ( E ) , ℘ ( x ) , ℘ c ( x c ))= tr . d . ( j ( E ) , ℘ ( xx c )) . By the wESC K ,lin . d . K ( xx c ) + tr . d . ( j ( E ) , ℘ ( x ) , ℘ ( x c )) − lin . d . Q ( xx c / Λ) ≥ − tr . d . ( K ) . Combining the above inequalities we obtain,2 tr . d . ( j ( E ) , ℘ ( x )) ≥ lin . d . Q ( xx c / Λ) − lin . d . K ( xx c ) − tr . d . ( K ) . Now, on the one hand, since ǫ is not in R ∪ i R , we know ǫ and ǫ c are R -linearlyindependent and hence lin . d . Q ( xx c ) = lin . d . Q ( x ) + lin . d . Q ( x c ) = 2 n . Thereforelin . d . Q ( xx c / Λ) ≥ lin . d . Q ( xx c ) − lin . d . Q (Λ) = 2 n − . On the other hand, since x c = ǫ c ǫ x and ǫ c ǫ ∈ K , we havelin . d . K ( xx c ) ≤ lin . d . K ( x ) ≤ n. Thus, 2 tr . d . ( j ( E ) , ℘ ( x )) ≥ (2 n − − n − tr . d . ( K ) = n − − tr . d . ( K ) . Hence, using the additivity properties of the transcendence degree, we see that2 tr . d . ( ℘ ( x )) = 2 tr . d . ( j ( E ) , ℘ ( x )) − . d . ( j ( E ) /℘ ( x )) ≥ . d . ( j ( E ) , ℘ ( x )) − ≥ n − − tr . d . ( K )and, similarly, 2 tr . d . k ( ℘ ( x )) = 2 tr . d . ( ℘ ( x )) − . d . ( k /℘ ( x )) ≥ n − − tr . d . ( K ) − . d . ( k ) . Because ℘ ( x ) and y = exp E ( x ) are interalgebraic over k , we obtain the inequality2 tr . d . k ( y ) ≥ n − − tr . d . ( K ) − . d . ( k ). (cid:3) By the same argument as in Section 4, one derives the following proposition.
Proposition 5.4.
For every tuple c ⊂ E , there exists a tuple c ′ ⊂ E , extending c ,such that c ′ is strong with respect to the predimension function ( δ G ) c ′ . If c ⊂ G ,then we can find such a c ′ also contained in G . Existential Closedness.
The following proposition completes the proof ofTheorem 5.2.
Proposition 5.5.
The structure ( A, G ) has the EC-property. Therefore, for everytuple c ⊂ G , strong with respect to ( δ G ) c , the structure ( A, G ) X is a model of thetheory T , where X = span( c ) with the structure induced from ( A, G ) . The proof of the above proposition is the same as in Section 4, with only verysmall differences. In order to be explicit about the differences, we review the differentsteps of the proof.For the rest of Subsection 5.3, let us fix an even number n ≥ V ⊂ ( C ∗ ) n of dimension n defined over k ( C ) for some finite subset C of E .We need to show that the intersection V ∩ G n is Zariski dense in V .Let us define the set X = { ( s, t ) ∈ R n : exp E ( ǫt + s ) ∈ V } . Note that if ( s, t ) is in
X ∩ (( Z + α Z ) n × R n ), where α = Re( τ ) − Re( ǫ )Im( ǫ ) Im( τ ), thenthe corresponding point y := exp E ( ǫt + s ) is in V ∩ G n (see 5.1). Thus, in orderto find points in the intersection V ∩ G n , we shall look for points ( s, t ) in X with s ∈ ( Z + α Z ) n .As in the previous section, our strategy is to find an implicit function for X defined on an open set S ⊂ R n , assigning to every s ∈ S a point t ( s ) ∈ R n suchthat ( s, t ( s )) ∈ X . Since ( Z + α Z ) n is dense in R n , the intersection S ∩ ( Z + α Z ) n is non-empty, and therefore we can find points ( s, t ( s )) in X with s ∈ ( Z + α Z ) n .Let R be an o-minimal expansion of the real ordered field in a countable languagein which the function ℘ is locally definable (and therefore also the set X ) and havingconstants for the real and imaginary parts of each element of k ( C ). The existenceof such a structure R , as a reduct of R an , follows from the fact that the additionformula , ℘ ( z + z ) = − ℘ ( z ) − ℘ ( z ) + 14 ( ℘ ′ ( z ) − ℘ ′ ( z ) ℘ ( z ) − ℘ ( z ) ) , allows to locally define ℘ in terms of its restriction to a closed parallelogram con-tained in the interior of the fundamental parallelogram of vertices 0 , , τ, τ (e.g.the one with vertices τ , τ , τ , τ ), around which it is analytic ([Mac05]).Indeed, this corresponds to the fact that exp E is a homomorphism and its valuescan therefore be calculated from those of any restriction to an open subset of thefundamental parallelogram.The proof of Proposition 5.5 relies on the following main lemma: Lemma 5.6 ( Main Lemma ) . Suppose ( s , t ) is an R -generic point of X , i.e. dim R ( s , t ) = dim R X = n . Then dim R ( s ) = n . To prove the Main Lemma, define the following set:
Definition 5.7.
For s ∈ C n we define the set L s = { ( x, ¯ x ) ∈ C n : ( x + ¯ x ) + β − i ( x − ¯ x ) = 2 s } . With the same proof as in Section 4, we have:
Lemma 5.8.
Suppose s ∈ R n . Then for all linearly independent ( m , n ) , . . . , ( m k , n k ) ∈ Z n ( m i , n i ∈ Z n ), we have dim( m, n ) · L s ≥ k . ATURAL MODELS OF THEORIES OF GREEN POINTS 25
Lemma 5.9.
Let s ∈ R n . Then the pair ( L s , V × V c ) is K -rotund. The proof of the Main Lemma of Section 4 from the analogous lemmas (see theend of Subsection 4.4) also works word by word in the new case.The next lemma follows from the Main Lemma by the same argument as inSection 4.
Lemma 5.10.
Suppose ( s , t ) is an R -generic point of X . There is a continuous R -definable function s t ( s ) defined on a neighbourhood S ⊂ R n of s and takingvalues in R n such that for all s ∈ S , the point y ( s ) := exp E ( ǫt ( s ) + s ) is in V . Finally, also the proof of Proposition 5.5 from the lemma above is the same asthe corresponding proof in Section 4, this time using the density of Z + α Z in R ,instead of that of the subgroup Q .The question of whether the model on E is ω -saturated is open. Let us simplynote that the remarks in Subsection 4.5 can be easily adapted to the elliptic curvecase case. References [Ax71] James Ax. On Schanuel’s conjectures.
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