A nondefinability result for expansions of the ordered real field by the Weierstrass ℘ function
AA NONDEFINABILITY RESULT FOR EXPANSIONS OF THEFIELD OF REAL NUMBERS BY THE RESTRICTEDWEIERSTRASS ℘ FUNCTION
RAYMOND MCCULLOCH
Abstract.
Suppose that Ω is a complex lattice that is closed under complexconjugation and that I is a small real interval, and that D is a disc in C . Then therestriction ℘ | D is definable in the structure ( ¯ R , ℘ | I ) if and only if the lattice Ω hascomplex multiplication. This characterises lattices with complex multiplication interms of definability. Introduction
Since the 1980’s model theorists have studied expansions of the real ordered field¯ R = ( R , <, + , · , ,
1) by various analytic functions. A major result in this area wasdue to Wilkie, who in [16] proved the model completeness of R exp = ( ¯ R , exp) whereexp is the real exponential function and this lead to much further activity.In particular in [3] Bianconi used Wilkie’s ideas together with a functional transcen-dence result due to Ax to show that no arc of the sine function is definable in R exp . If C is identified with R , this result may be rephrased to say that no restriction of thecomplex exponential function to an open disc in C is definable in R exp . Bianconi wentfurther in [4] and showed that if f : D → C is holomorphic and definable in R exp then f is algebraic. This result is used by Peterzil and Starchenko in [13] to characteriseall definable locally analytic subsets of C n in the structure R exp .Formulated in this way, the question can be generalised to other functions. Therecertainly are transcendental functions f such that there are discs D ⊆ C with f | D definable in the structure ( ¯ R , f | I ) for some interval I ⊆ R . Indeed it turns out thatexamples occur with functions not so different from exp. Recall that to a latticeΩ ⊆ C (i.e a discrete subgroup of rank 2) Weierstrass associated a meromorphicfunction, ℘ ( z ) = ℘ Ω ( z ) = 1 z + (cid:88) ω ∈ Ω ∗ (cid:18) z − ω ) − ω (cid:19) , where Ω ∗ = Ω \ { } . This meromorphic function ℘ has poles at exactly the points inΩ and is periodic with respect to Ω. In Section 2 we shall state the addition formulafor ℘ and its differential equation. We can see that ℘ is similar to exp as they areboth periodic and have an addition formula and a differential equation. They are alsosimilar as they both give an exponential map of a commutative algebraic group.In the course of his investigations into the model theory of these elliptic functions,Macintyre observed in [12] that if Ω = Z + i Z then in the structure ( ¯ R , ℘ | (1 / , / )the restriction of ℘ to any disc D on which ℘ is analytic is definable. (The interval(1/8,3/8) is chosen for convenience as it avoids both poles of ℘ and the zeros of ℘ (cid:48) .Any such interval may be chosen.)However the lattice Ω = Z + i Z is rather special. It can easily be seen that ℘ ( iz ) = − ℘ ( z ). This formula is all that is required to prove the aforementioned a r X i v : . [ m a t h . L O ] J u l RAYMOND MCCULLOCH observation of Macintyre in [12]. In particular there are non-integer α ∈ C suchthat α Ω ⊆ Ω. (Any α ∈ i Z for example). Such lattices are said to have complexmultiplication . The lattice Ω = Z + i Z is also closed under complex conjugation inother words ¯Ω = Ω. Lattices satisfying this latter property are known as real lattices .It can be seen in Section 18 of [15] that if Ω is a real lattice then ℘ is real valued whenrestricted to an interval on either the real or imaginary axis. Hence such restrictionsof ℘ are real analytic functions and it is natural to consider the model theory of theserestrictions. In Section 3 it is shown that Macintyre’s observation extends easily to allreal lattices with complex multiplication. It can then be shown that this characterisesthose Ω for which ℘ | I for some finite real interval I defines ℘ | D for some disc D . Thisis done in the following theorem. Theorem 1.1.
Let Ω be a real lattice and let ℘ Ω be its ℘ function. Let I be somereal interval not containing a pole. Then there’s a non-empty disc D ⊆ C such that ℘ | D is definable in the structure ( ¯ R , ℘ | I ) if and only if there’s an α ∈ C \ Z such that α Ω ⊆ Ω . In Section 5 we give the proof of Theorem 1.1. The theorem is proved by adaptingthe method of Bianconi in [3]. In [9] Jones, Kirby and Servi attempt to apply thismethod to answer questions about the local interdefinability of Weierstrass ℘ functions.However it turns out this method cannot be applied to their problem and they turnto the method of predimensions of Hrushovski in [8]. In fact their method cannot beused here. The method of Bianconi involves using a theorem of Wilkie on smoothfunctions that are defined implicitly that was proved generally by Jones and Wilkie in[10]. Bianconi refers to these methods of Wilkie as the “Desingularisation Theorem”.We shall obtain this implicit definition in Section 4 and also explain why this implicitdefinition may be used to prove Theorem 1.1.2. Background on the Weierstrass ℘ function Here we give all the background that shall be needed on the Weierstrass ℘ function.All of this can be found in the books [6] or [14]. The function ℘ is analytic except atits poles which are at exactly the points in the lattice Ω. Also ℘ is doubly periodicand Ω is its period lattice. Clearly ℘ depends on Ω. As mentioned in the introductionto this paper ℘ has an addition formula, which is now stated. Let z, w be complexnumbers such that z − w / ∈ Ω. Then, ℘ ( z + w ) = 14 (cid:18) ℘ (cid:48) ( z ) − ℘ (cid:48) ( w ) ℘ ( z ) − ℘ ( w ) (cid:19) − ℘ ( z ) − ℘ ( w ) . (2.1)This gives rise to the duplication formula, ℘ (2 z ) = 14 (cid:18) ℘ (cid:48)(cid:48) ( z ) ℘ (cid:48) ( z ) (cid:19) − ℘ ( z ) . (2.2)Another important property of ℘ is that it satisfies a differential equation namely( ℘ (cid:48) ( z )) = 4 ℘ ( z ) − g ℘ ( z ) − g , (2.3)where the complex numbers g and g are called the invariants of ℘ . From thisdifferential equation it is clear that there is an algebraic relation between ℘ and ℘ (cid:48) .Differentiating both sides of this equation gives us that ℘ (cid:48)(cid:48) ( z ) = 6 ℘ ( z ) − g . (2.4) NONDEFINABILITY RESULT FOR WEIERSTRASS ℘ FUNCTIONS 3 An elliptic function f is a meromorphic function on C which has two complexperiods ω and ω (cid:48) such that Im ( ω (cid:48) /ω ) >
0. These periods generate a lattice of periodsΩ. If we denote the field of elliptic functions for a fixed period lattice Ω by L thenit is known that this field L is in fact C ( ℘, ℘ (cid:48) ). This can be seen in Theorem 3.2 inChapter 6 of [14]. Remark . From Section 2 of [15] it can be seen that real lattices may be split intotwo cases which are known as the rectangular and rhombic lattices. A lattice Ω iscalled a rectangular lattice if we can choose generators ω and ω of Ω such that ω isreal and ω is purely imaginary. The lattice Ω is called a rhombic lattice if we canchoose generators of Ω, namely ω and ω such that ω = ω .In Section 5 we give the proof of Theorem 1.1 in the rectangular lattice case, the proofof the rhombic case is similar.Finally we conclude this background section by stating a version of Ax’s theoremfor the Weierstrass ℘ function. This is a theorem of Ax in [1] and Brownawell andKubota in [5]. Theorem 2.2.
Let Ω be a complex lattice which does not have complex multiplication.Let z , . . . , z n be power series with complex coefficients and no constant term that arelinearly independent over Q . Then we have thattr.deg C C [ z , . . . , z n , ℘ ( z ) , . . . , ℘ ( z n )] ≥ n + 1 . Macintyre’s Result
Recall that if a complex lattice Ω has complex multiplication then there’s a non-integer α such that α Ω ⊆ Ω. Lemma 3.1.
Let Ω be a real lattice that is closed under complex conjugation and let ℘ Ω be its ℘ function. Let I be some real interval not containing a pole. Then therestriction of ℘ to any complex disc not meeting any poles is definable in the structure ( ¯ R , ℘ | I ) .Proof. We follow the proof of Macintyre in [12] for the case Ω = Z + i Z . This justchecks the proof works in the general case. Using the extra symmetry of the lattice Ω,due to complex multiplication, we now show that ℘ restricted to αI , where α ∈ C \ Z issuch that α Ω ⊆ Ω, is definable in the structure ( ¯ R , ℘ | I ). Let z ∈ I and f ( z ) = ℘ ( αz ) . Then for any ω ∈ Ω, we can see that, f ( z + ω ) = ℘ ( αz + αω ) = ℘ ( αz ) = f ( z )as α Ω ⊆ Ω. Therefore f is a meromorphic function that is doubly periodic with respectto the lattice Ω. So f is an elliptic function for the lattice Ω. Hence f may be writtenas a rational function R in ℘ ( z ) and ℘ (cid:48) ( z ). Similarly the function g ( z ) = ℘ (cid:48) ( αz ) maybe written as a rational function S in ℘ ( z ) and ℘ (cid:48) ( z ).Therefore both the functions ℘ and ℘ (cid:48) restricted to αI are definable in the structure( ¯ R , ℘ | I ). Now consider some disc D contained in I × αI . For z ∈ D it is clear thatwe may write z = x + αy for x, y ∈ I . We can assume that x − αy / ∈ Ω . Then by theaddition formula, ℘ ( z ) = ℘ ( x + αy ) = 14 (cid:18) ℘ (cid:48) ( x ) − S ( ℘ ( y ) , ℘ (cid:48) ( y )) ℘ ( x ) − R ( ℘ ( y ) , ℘ (cid:48) ( y )) (cid:19) − ℘ ( x ) − R ( ℘ ( y ) , ℘ (cid:48) ( y )) . RAYMOND MCCULLOCH
As every function in this expression is definable we have that the function ℘ | D isdefinable in the structure ( ¯ R , ℘ | I ). Using the addition formula gives us that ℘ restrictedto any disc in C is definable in ( ¯ R , ℘ | I ) as required. (cid:3) Getting an implicit definition
In this section we shall describe the implicit definition that shall be needed in theproof of Theorem 1.1. This implicit definition is due to Wilkie in [16] and is referredto by Bianconi as the ”Desingularisation Theorem” and was proved in a more generalform by Jones and Wilkie in [10]. We let ˜ R = ( ¯ R , F ) be an expansion of ¯ R by aset F of total analytic functions in one variable, closed under differentiation that iso-minimal with a model complete theory. Desingularisation can be used to show thatif the function f : U → R , for some U ⊆ R , is a definable function of ˜ R then f maybe defined piecewise by a system of equations whose matrix of partial derivativesis non-singular. More precisely there’s some interval I (cid:48) ⊆ U , some integer n ≥ , and there are certain functions F , . . . , F n : R n +1 → R (see below) and functions f , . . . , f n : I (cid:48) → R such that for all t ∈ I (cid:48) , F ( t, f ( t ) , f ( t ) , . . . , f n ( t )) = 0... F n ( t, f ( t ) , f ( t ) , . . . , f n ( t )) = 0and det (cid:18) ∂F i ∂x j (cid:19) i =1 ,...,nj =2 ,...,n +1 ( t, f ( t ) , f ( t ) , . . . , f n ( t )) (cid:54) = 0 . Here the functions F , . . . , F n are polynomials in x , . . . , x n +1 and g i ( x j ) for i =1 , . . . , l and j = 1 , . . . , n + 1 where g , . . . , g l ∈ F .To apply this to our setting we can see from (2.4) that there is a polynomialexpression for ℘ (cid:48)(cid:48) in terms of ℘ and its invariants. Using this expression and thedifferential equation (2.3) for ℘ we may find similar expressions in ℘ and ℘ (cid:48) for everyderivative of ℘. Hence we have that the ring of terms with parameters from R ofthe structure ( ¯ R , ℘ | I ) is closed under differentiation. The structure ( ¯ R , ℘ | I ) is modelcomplete by a theorem of Gabrielov in [7]. Bianconi also has model completenessresults involving the ℘ function in [2] however these seem difficult to apply here asthey are for the complex functions rather than their restrictions to a real interval.The main obstacle to using this implicit definition in the proof of Theorem 1.1 isthat the function ℘ | I is not a total function on R . In order to use this method thesefunctions are made to be total functions on R . We do this by composing ℘ with abijection from R to the interval I . This bijection is defined for a general interval I = ( a, b ) here. In the proof of Theorem 1.1 we give an explicit interval but do notexplicitly define the bijection. Firstly define A : ( a, b ) → R by A ( t ) = t − b + a (cid:0) b − a (cid:1) − (cid:0) t − b + a (cid:1) , which is a bijection. Differentiating gives, A (cid:48) ( t ) = (cid:0) b − a (cid:1) + (cid:0) t − b + a (cid:1) (cid:16)(cid:0) b − a (cid:1) − (cid:0) t − b + a (cid:1) (cid:17) , NONDEFINABILITY RESULT FOR WEIERSTRASS ℘ FUNCTIONS 5 which clearly doesn’t vanish. The inverse, B = A − is also differentiable and B (cid:48) ( t ) = (cid:16)(cid:0) b − a (cid:1) − (cid:0) B ( t ) − b + a (cid:1) (cid:17) (cid:0) b − a (cid:1) + (cid:0) B ( t ) − b + a (cid:1) also doesn’t vanish. Finally we define B ( t ) = 1 (cid:0) b − a (cid:1) + (cid:0) B ( t ) − b + a (cid:1) . Therefore the structure ( ¯ R , ℘ ◦ B, B, B ) is an expansion of ¯ R by total analyticfunctions on R . The structures ( ¯ R , ℘ | I ) and ( ¯ R , ℘ ◦ B, B, B ) are equivalent in thesense that they have the same definable sets. Hence the structure ( ¯ R , ℘ ◦ B, B, B ) isalso model complete and has a ring of terms with parameters from R that is closedunder differentiation. Therefore in the proof of Theorem 1.1 it is sufficient to passfrom the structure ( ¯ R , ℘ | I ) to the auxiliary structure ( ¯ R , ℘ ◦ B, B, B ) and prove thetheorem in this structure. In the next section it is explained how we may use thisimplicit definition in the structure ( ¯ R , ℘ ◦ B, B, B ) in order to prove Theorem 1.1.5. Proof of Theorem 1.1
In this section we give the proof of Theorem 1.1. The argument uses a method ofBianconi in [3] as well as [4], which involves finding upper and lower bounds on thetranscendence degree of some finite extension of C that are contradictory. The lowerbound is found using Theorem 2.2 and the upper bound requires a similar argumentto that used to prove Claim 4 in the proof of Theorem 4 in [4]. Proof of Theorem 1.1.
By Remark 2.1 we know that real lattices may be divided intotwo cases, the rectangular and rhombic lattices. Firstly we note that one direction ofTheorem 1.1 is proved in Lemma 3.1. Here we prove the other direction of Theorem1.1 for the rectangular lattice case as both cases are fairly similar. The main differenceis the choice of generators of Ω that may be made and the intervals that can be chosenaccordingly. The full details of both cases will appear in my PhD thesis.Let Ω be a rectangular real lattice. From Section 19 of [15] we can see that we canchoose generators for the lattice Ω, namely ω and ω such that ω is real and ω ispurely imaginary. This leads to two cases, when | ω | ≤ | ω | and when | ω | ≤ | ω | .We shall want to find a real interval I not containing any lattice points such that iI also doesn’t contain any lattice points. In the first case the interval ( ω / , ω / ω / i, ω / i ) is chosen. It suffices toprove the theorem in either case and so we assume we are in the first of these casesand that I = ( ω / , ω / R , ℘ ◦ B, B, B ) andwe therefore now pass to this structure.Now we assume for a contradiction that Ω doesn’t have complex multiplication andthat there’s a non-empty disc D ⊆ C such that the restriction ℘ | D is definable in thestructure ( ¯ R , ℘ ◦ B, B, B ). By translating and scaling using the addition formula(2.1) we may take D to contain the interval iI = ( iω / , iω / | ω | ≤ | ω | theinterval iI doesn’t contain any lattice points. Hence the function f : I → R definedby f ( t ) = ℘ ( iB ( t )) is definable in the structure ( ¯ R , ℘ ◦ B, B, B ). By the discussionin Section 4 there’s an open interval I (cid:48) ⊆ ( ω / , ω / n ≥ , certain RAYMOND MCCULLOCH functions F , . . . , F n : R n +1 → R and functions f , . . . , f n : I (cid:48) → R such that for all t ∈ I (cid:48) , F ( t, f , . . . , f n ) = 0... F n ( t, f , . . . , f n ) = 0and det (cid:18) ∂F i ∂x j (cid:19) i =1 ,...,nj =2 ,...,n +1 ( t, f , . . . , f n ) (cid:54) = 0 . The functions F , . . . , F n are polynomials in x , . . . , x n +1 , ℘ ( B ( x )) , . . . , ℘ ( B ( x n +1 )) ,B ( x ) , . . . , B ( x n +1 ) , B ( x ) , . . . , B ( x n +1 ) . However as B and B are algebraic and ℘ and ℘ (cid:48) are algebraically dependent we maytake the functions F , . . . , F n to be algebraic in x , . . . , x n +1 and ℘ ( B ( x )) , . . . , ℘ ( B ( x n +1 ))after potentially moving to a smaller interval.Take n to be minimal such that there exists an interval I (cid:48) and functions F , . . . , F n in x , . . . , x n +1 and ℘ ( B ( x )) , . . . , ℘ ( B ( x n +1 )) such that the above system of equationsand non-singularity condition holds. As we are restricted to a small interval it ispossible to shrink this interval to avoid the zeros of ℘ (cid:48) if necessary. In fact the intervalmay be shrunk repeatedly if needed.The desired contradiction will be obtained by considering upper and lower boundson the transcendence degree of a certain finite extension of C . These bounds will beincompatible.Firstly we find an lower bound on transcendence degree. In order to find thislower bound we want to apply Theorem 2.2. To apply this theorem it must firstbe shown that B ( t ) − B (0) , B ( f ( t )) − B ( f (0)) , . . . , B ( f n ( t )) − B ( f n (0)) are linearlyindependent over Q . As the lattice Ω doesn’t have complex multiplication this linearindependence is only required over Q . This linear independence is shown by showing that linear dependence contradicts theminimality of n . This linear independence argument does not depend on whether thelattice Ω is rectangular or rhombic. If B ( t ) − B (0) , B ( f ( t )) − B ( f (0)) , . . . , B ( f n ( t )) − B ( f n (0)) are assumed to be linearly dependent over Q then it can be shown that B ( f n ( t )) − B ( f n (0)) can be written as a Q -linear combination of B ( t ) − B (0) , B ( f ( t )) − B ( f (0)) , . . . , B ( f n − ( t )) − B ( f n − (0)). For any natural number n it is possible toobtain a formula ℘ ( nz ) in terms of ℘ ( z ) and ℘ (cid:48) ( z ) using the addition and duplicationformulas (2.1) and (2.2). Similarly we can write ℘ ( z ) as an algebraic expression in ℘ ( z/n ) and ℘ (cid:48) ( z/n ). Hence there’s an algebraic expression for ℘ ( z/n ) in terms of ℘ ( z ) and ℘ (cid:48) ( z ). Therefore the functions F , . . . , F n may be rewritten as functions in t, f , . . . , f n − . This gives a new system of equations in n variables. If this new systemhas a matrix of partial derivatives that’s non-zero at ( t, f , . . . , f n − ) then there’sa contradiction to the minimality of n . Hence it is assumed that all the minors ofthe matrix of partial derivatives for this system have determinant equal to zero andfrom this assumption a contradiction to the non-singularity of the original system isobtained.Now we observe that adding i ( B ( t ) − B (0)) to the list B ( t ) − B (0) , B ( f ( t )) − B ( f (0)) , . . . , B ( f n ( t )) − B ( f n (0)) doesn’t destroy this linear independence. For if it NONDEFINABILITY RESULT FOR WEIERSTRASS ℘ FUNCTIONS 7 did then we could write for rational a , . . . , a n not all zero, i ( B ( t ) − B (0)) = a ( B ( t ) − B (0))+ a ( B ( f ( t )) − B ( f (0)))+ · · · + a n ( B ( f n ( t )) − B ( f n (0))) . The left hand side of this expression is non-real whereas the right hand side isreal for all t in the interval I (cid:48) , a contradiction. Now applying Theorem 2.2 to iB ( t ) , B ( t ) , B ( f ( t )) , . . . , B ( f n ( t )) gives thattr.deg C C [ iB ( t ) , B ( t ) , B ( f ( t )) , . . . , B ( f n ( t )) , ℘ ( iB ( t )) ,℘ ( B ( t )) , ℘ ( B ( f ( t ))) , . . . , ℘ ( B ( f n ( t )))] ≥ n + 3 . Now we obtain a upper bound on the transcendence degree of the same finite extensionof C that is smaller than this lower bound. Recall that on the interval I (cid:48) the function f ( t ) = ℘ ( iB ( t )) is defined by a system of equations F , . . . , F n in the variables x , . . . , x n +1 , whose Jacobian is non-singular. For each i = 1 , . . . , n the function F i can be written as F i ( t, f , . . . , f n ) = P i ( t, f , . . . , f n , ℘ ( B ( t )) , ℘ ( B ( f ( t ))) , . . . , ℘ ( B ( f n ( t )))) , where P , . . . , P n are algebraic functions in y , . . . , y n +2 . This is a system of n equations in 2 n + 2 variables. We check that the matrix (cid:18) ∂P i ∂y j (cid:19) i =1 ,...,nj =2 ,..., n +2 ( t, f , . . . , f n , ℘ ( B ( t )) , ℘ ( B ( f ( t ))) , . . . , ℘ ( B ( f n ( t ))))is of maximal rank n . This is done by a similar argument to that of Claim 4 in theproof of Theorem 4 in [4]. Differentiating F i with respect to x j gives ∂F i ∂x j = ∂P i ∂y j + B (cid:48) ( x j ) ℘ (cid:48) ( B ( x j )) ∂P i ∂y j + n +1 . Therefore the matrix ( ∂F i /∂x j ) is given by multiplying the matrix ( ∂P i /∂y j ) by a(2 n + 1) × n matrix M .Here M is the matrix whose first n × n block is the identity matrix followed by a row ofzeros and finally an n × n diagonal matrix diag( B (cid:48) ( x ) ℘ (cid:48) ( B ( x )) , . . . , B (cid:48) ( x n +1 ) ℘ (cid:48) ( B ( x n +1 ))) . In other words M is the matrix M = . . . . . . . . . B (cid:48) ( x ) ℘ (cid:48) ( B ( x )) . . . . . . B (cid:48) ( x n +1 ) ℘ (cid:48) ( B ( x n +1 )) . By the non-singularity of the system F , . . . , F n it’s clear that the rows of the matrix( ∂F i /∂x j ) are linearly independent over R and so the rows of the matrix ( ∂P i /∂y j )are also linearly independent over R . So this matrix is of maximal rank n as required.Thereforetr.deg C C [ t, f ( t ) , . . . , f n ( t ) , ℘ ( B ( t )) , ℘ ( B ( f )) , . . . , ℘ ( B ( f n ))] ≤ n + 2 . However this is an upper bound on the transcendence degree on a slightly differentextension of C . But iB ( t ) and B ( t ) are algebraically dependent as are B ( ℘ ( iB ( t ))) RAYMOND MCCULLOCH and ℘ ( iB ( t )) and f i and B ( f i ) for all i = 1 , . . . , n as B is algebraic. Therefore wehave thattr.deg C C [ iB ( t ) , ℘ ( iB ( t )) , B ( t ) , ℘ ( B ( t )) , B ( f ( t )) , . . . , B ( f n ( t )) ,℘ ( B ( f ( t ))) , . . . , ℘ ( B ( f n ( t )))] ≤ n + 2 . So we have found upper and lower bounds on the transcendence degree of some finiteextension of C that are incompatible. 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