A Schanuel Property for $j$

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by  Sebastian Eterović

A SCHANUEL PROPERTY FOR j
SEBASTIANETEROVIĆ
Abstract. I give a model-theoretic setting for the modular j function and its derivatives. These
structures, here called j-fields, provide an adequate setting for interpreting the Ax-Schanuel theorem
7 for j (Theorem 1.3 of [14]). Following the ideas of [3] and [6] for exponential fields, I prove a generic
1 transcendence property forthej function.
0
2
1. Introduction

n
a The aim of this work is to prove a generic transcendence property for the j function in the spirit of
J Schanuel’sconjecture(seeconjecture(S)of[2]). Sucharesultisprovedin[3]fortheexponentialfunction

0 in the context of exponential fields.
2
Theorem 1.1 (see Theorem 1.2 of [3]). Let F be any exponential field, let λ F be exponentially
∈
] transcendental, and let x Fn be such that exp(x) is multiplicatively independent. Then:
O ∈
t.d.(exp(x),exp(λx)/λ) n.
L
≥
h. The notation we use is as follows: for A and B subsets of a given field of characteristic zero, we write
t t.d.(A/B) to denote the transcendence degree of the field extension Q(A)/Q(B). Given some elements
a
m z1,...,zn in the field F, we write z to denote the tuple (z1,...,zn), and if f is a function, then f(z)
denotes the tuple (f(z ),...,f(z )).
[ 1 n
ThisresultreliesheavilyonTheorem3of[2](theso-calledAx-Schanueltheorem). In[14],theauthors

1 prove an analogue of the Ax-Schanuel theorem for the j function (see Theorem 6.1), so it is natural to
v wonder if a similar strategy would provide a transcendence result for j. This is what is done here.
1
To do this we willintroduce the notionofj-fields(in analogyto the wayexponentialfields aredefined
4
in [6]). We will use some standard techniques of geometric model theory (like pregeometries) and o-
8
5 minimality for our approach. A very short introduction to pregeometries is given in the Preliminaries.
0 The proofof our main relies mostly ontranscendence properties of fields, so the readernot familiar with
1. the model-theoretic concepts may still follow the main strategy.
0 Traditionally,the j function is understoodas a modular function defined onthe upper-half plane, but
7 we will extend it to be defined on the upper and lower half planes so that j : H+ H− C. Given
1 z H− we define j(z) := j(z), where z is the complex conjugate of z (even though∪the s→ymbol is the
: ∈
v same,itshouldbeclearfromcontextwhenweusez todenotecomplexconjugateoratupleofelements).
i We do this because the condition for an element g GL (Q) to preserve H+ H− (setwise) is easier to
X 2
∈ ∪
check: namely det(g) = 0. The importance of this will become apparent in the proof of Lemma 4.1. If
r 6
a we considered only the upper-half plane, we would have to ask that det(g) > 0, which would require us
to include an order relation in our j-field. Given that we are interested in transcendence properties of j,
then it does not matter if we extend the domain of j in the way we have done.
The term j-generic used in the next theorem is analogous to that of exponential transcendence; we
will give a precise definition in Section 5. The full statement of our main result (Theorem 6.9) is rather
technical and works on all j-fields, but one immediate consequence to a concrete setting is:
Theorem 1.2 (see Corollary 6.4). Let τ R be j-generic. Suppose z ,...,z H+ H− and g
1 n
∈ ∈ ∪ ∈
GL (Q(τ)) are such that z ,...,z ,gz ,...,gz are in different GL (Q)-orbits (pairwise). Then:
2 1 n 1 n 2
t.d.(j(z),j′(z),j′′(z),j(gz),j′(gz),j′′(gz)/τ) 3n.
≥
Loosely, this should be understood as: if the entries of g are sufficiently generic (transcendental with
respecttoj),thenweobtainthe abovetranscendenceresultforj andits derivatives. Usingo-minimality
we prove that there are uncountably many values for τ that can be used (see Remark 4), although, just
Date:January23,2017.
1
2 SEBASTIANETEROVIĆ
as we do not have any explicit examples of exponentially transcendental numbers, no explicit j-generic
numbers are known.
As will be shown, j-fields prove to be an adequate setting in which to interpret the Ax-Schanuel
theoremfor j (see Theorem6.1), that is to say,we can define certainpregeometries(here named gcl and
jcl) on j-fields that encode the content of this result.
Thesectionsareorganizedasfollows. Firstwedefinej-fieldsaxiomaticallysoastomimicthebehaviour
of the j function on C. Then we study the family of pregeometries gcl obtained from the action of a
F
subgroupofGL (F)onafieldK,whereF isasubfieldofK. Wealsointroducethe notionof“geodesical
2
disjointness”,whichis aconceptthatemulatesthe ideaoftwofields being “linearlydisjoint” overa third.
Lemmas 4.1 and 4.2 are crucial for obtaining our main theorem.
Afterthat,comesasectiononj-derivations,whicharederivationswhichrespectthej functionandits
derivatives. We define the operator jcl and prove that it is a pregeometry. Using standard techniques of
o-minimality,wealsoshowhowtoconstructnon-trivialj-derivationsonC. See[8]foraquickintroduction
to o-minimality.
The last section contains the transcendence results, the most important being Theorem 6.9 which
generalises Theorem 1.2. As said above, the strategy of the proofs is inspired by [3]. We also analyse
a modular version of Schanuel’s conjecture and prove that, like in the case of the original Schanuel’s
conjecture, in C there are at most countably many essential counterexamples of this conjecture. At the
endof this sectionwe presentsome openproblems. As is explained there,because of the abstractnature
ofderivations,it is not easyto decide when anelement does not belongs to the jcl-closureof a givenset,
and a more explicit definition of jcl would be helpful. In the case of exponential fields, one can do this
through Khovanskii systems (see Proposition 7.1 of [6]). However, the strategy used there relies on very
specific properties of the exponential function, and so does not work for jcl.
2. Preliminaries
2.1. Pregeometries. Let X be a set. A function cl: (X) (X) (here (X) denotes the power set
P →P P
of X) is called a pregeometry on X if it satisfies the following properties for every A,B (X):
∈P
(a) A cl(A),
⊆
(b) If A B, then cl(A) cl(B),
⊆ ⊆
(c) cl(cl(A))=cl(A),
(d) Finite character: if a cl(A), then there is a finite subset A A such that a cl(A ),
0 0
∈ ⊆ ∈
(e) Exchange: if a,b X are such that a cl(A b ) and a / cl(A), then b cl(A a ).
∈ ∈ ∪{ } ∈ ∈ ∪{ }
For example, if X is a vector space, then we can take cl(A) to be the linear span of A. If X is a field,
then we can take cl(A) to be the set of all elements of X algebraic over A.
A crucial aspect of pregeometries is that they allow us to have well-defined notions of independence
and dimension. A set A X is cl-independent if for every a A we have that a / cl(A a ). The
⊆ ∈ ∈ \{ }
dimension ofasetB X with respecttocl is the cardinalityofanycl-independentset A B suchthat
⊆ ⊆
cl(A)=B. For more basic properties of pregeometries, see Appendix C of [18].
2.2. Review of Elliptic Curves and the Modular j function. We will nowgive a verybrief review
of complex elliptic curves only to have enough context for the set up of the j function. For the general
theory of elliptic curves see e.g. [16] and [17].
A lattice in C is a subgroup Λ of the additive group of C generatedby two elements ω ,ω C which
1 2
∈
are linearly independent over R. This means that Λ is isomorphic to Zω +Zω . The quotient space
1 2
E = C/Λ has a natural complex structure induced by that of C. In fact E is a compact Riemann
Λ Λ
surfaceofgenus1 andit canbe realisedasa complex plane projectivecurveby meansofthe Weierstrass
℘ associated to Λ as follows.
Given the lattice Λ, consider the Weierstrass ℘ function:
1 1 1
℘(z)= + .
z2 (cid:18)(z λ)2 − λ2(cid:19)
λ∈XΛ,λ6=0 −
UsingtheWeierstrassM-testforconvergence,onecanseethat℘(z)isameromorphicfunctiononCthat
is Λ-invariant. It also satisfies the differential equation: ℘′(z)2 =4℘(z)3 g ℘(z) g , where:
2 3
− −
1 1
g =60 , g =140 .
2 λ4 3 λ6
λ∈XΛ,λ6=0 λ∈XΛ,λ6=0
A SCHANUEL PROPERTY FOR j 3
So, the map [z] [℘(z) : ℘′(z) : 1] defines an isomorphism between E and the plane projective curve
Λ
7→
Y2Z =4X3 g XZ2 g Z3.
2 3
− −
Thisconstructionmotivatesthefollowingdefinition. Anelliptic curve isaplaneprojectivecurvegiven
by an equation of the form:
(2.1) Y2Z =4X3 aXZ2 bZ3,
− −
such that the discriminant ∆:=a3 27b2 =0. Any lattice Λ in C produces an elliptic curve E by the
Λ
− 6
above construction. In fact, all elliptic curves can be obtained in this way. Also notice that the analytic
constructionwe gave of E (as a quotient space C/Λ) immediately shows that E has a groupstructure
Λ Λ
(as a quotient group). But in fact the group operation is a rational map, and so it also compatible with
the algebraic structure of E (as a plane projective curve). For this reason we will use the name E to
Λ Λ
refer both to the analytic and algebraic aspects of the elliptic curve.
If E is an elliptic curve defined by equation (2.1), define the j-invariant of E as:
a3
j(E):=1728 .
∆
A morphism of elliptic curves is a morphism of algebraic varieties that is also a group homomorphism.
For any elliptic curve E′ we have that E is isomorphic to E′ if and only if j(E)=j(E′).
Given twolattices, Λ and Λ′, EΛ is isomorphicto EΛ′ if and only if there is c C∗ suchthat cΛ=Λ′.
∈
So,allisomorphismclassesofelliptic curvesareobtainedby consideringonlylattices ofthe formZ+Zτ,
where τ H. We can now formulate the definition we have been waiting for.
∈
Definition. The j-function is the holomorphic map j : H C such that j(τ) = j(E ), where E is
τ τ
→
the elliptic curve given by the lattice Λ(τ) := Z+Zτ. As was said in the introduction, we will extend
this function to be defined on the upper and lower half-planes, so that j :H+ H− C, where j(τ) for
∪ →
τ H− is defined as j(τ):=j(τ), where z denotes the complex conjugate of z.
∈
We will now review a few properties of this function. As it can be easily verified, given τ,τ′ H, we
∈
have that Λ(τ)=Λ(τ′) if and only if there is g SL (Z) such that:
2
∈
aτ +b a b
=τ′, where g = .
cτ +d (cid:18)c d(cid:19)
This means that j is invariant under the action of SL (Z). In fact, a lot more is known. The first
2
thing is to note that SL (R) acts on H through Möbius transformations, in fact SL (R) is the group of
2 2
automorphismsofH. Giveng GL+(Q)(the +denotespositivedeterminant),thereisauniquepositive
∈ 2
integer N(g) N such that N(g)g GL (Z) and the entries of N(g)g are relatively prime. There
2
∈ ∈
exists a family of polynomials Φ Z[X,Y], usually referred to as the modular polynomials, that
{ N}N≥1 ⊆
satisfy Φ (j(x),j(y)) = 0 if and only if there is g GL+(Q) such that det(N(g)g) = N and x = gy.
N ∈ 2
Φ (X,Y)=X Y andΦ (X,Y)issymmetricforN 2. Forapreciseconstructionofthesepolynomials,
1 N
− ≥
see e.g. [20].
So the modular polynomials give us algebraic properties of the j function. From the analytic side, it
is proven in [10] that j satisfies the following differential equation of order 3:
j′′′ 3 j′′ 2 j2 1968j+2654208
(2.2) + − (j′)2 =0.
j′ − 2(cid:18)j′(cid:19) (cid:18) 2j2(j 1728)2 (cid:19)
−
Because of this, for transcendence purposes of j and its derivatives, it suffices to consider only j, j′ and
j′′. Thanks to the Modular Ax-Lindemann-Weierstrass (ALW) theorem below we can also say that the
only algebraic relations that the functions j(z) and j(gz) can satisfy are those given by the modular
polynomials. On K, a field of characteristic zero, x ,...,x K are said to be geodesically independent
1 n
if there are no relations of the form x =gx , where i=k an∈d g GL+(Q).
i k 6 ∈ 2
Theorem 2.1 (Modular ALW, theorem 1.1 of [12]). Let C(W) be an algebraic function field, where
W Cn be an irreducible algebraic variety. Suppose that a ,...,a C(W) take values in H at some
1 n
⊆ ∈
P W. If a ,...,a are geodesically independent, then the 3n functions:
1 n
∈
j(a ),...,j(a ),j′(a ),...,j′(a ),j′′(a ),...,j′′(a )
1 n 1 n 1 n
(considered as functions on W locally near P) are algebraically independent over C(W).
The more general Ax-Schanuel theorem for j is stated later (see Theorem 6.1).
4 SEBASTIANETEROVIĆ
3. j-Fields
In this section we define j-fields. Given a field K, for any subfield F of K, there is a natural actionof
GL (F) on P1(K)=K , given by:
2
∪{∞}
ax+b
gx= ,
cx+d
a b
where g GL (F) is represented by g = . Whenever we say that GL (F) acts on K, it will be
∈ 2 (cid:18)c d(cid:19) 2
in this manner. Throughout, let G=GL (Q).
2
Definition. A j-field is a two-sorted structue K,D,α,j,j′,j′′,j′′′ , where:
h i
K= K,+, ,0,1 is a field of characteristic zero,
• h · i
D= D,(g) is a G-set (i.e. a set with an action of G),
• D g∈GE
and α,j,j′,j′′,j′′′ :D K are maps that satisfy:
→
(1) α is injective.
(2) For every z D and g G, α(gz)=gα(z).
∈ ∈
(3) For every z D,
∈
(j(z)=0 j(z)=1728 j′(z)=0) = k(j(z),j′(z),j′′(z),j′′′(z))=0,
6 ∧ 6 ∧ 6 ⇒
where k Q(X,Y,Z,W) is given by:
∈
W 3 Z 2 X2 1968X+2654208
(3.1) k(X,Y,Z,W):= + − Y2.
Y − 2(cid:18)Y (cid:19) (cid:18) 2X2(X 1728)2 (cid:19)
−
Thus k(j(z),j′(z),j′′(z),j′′′(z))=0 corresponds to the equation (2.2).
(4) Theaxiomscheme: foreveryz ,z D,ifz =gz ,thenΦ (j(z ),j(z ))=0(note thatgiveng
1 2 1 2 N 1 2
∈
wecanobtainthevalueofN asN =det(N(g)g),sothisaxiomschemeisfirst-orderexpressible).
Wealsoincludetheexpressionsthatcanbeobtainedbyderivingupto3-timesmodularrelations.
This meansthe following. We haveforeveryz D thatΦ (j(z),j(gz))=0. Ifwe interpretthis
N
∈
expression in C and derive it with respect to z, then we get:
ad bc
Φ (j(z),j(gz))j′(z)+Φ (j(z),j(gz))j′(gz) − =0,
N1 N2 (cz+d)2
where Φ and Φ are the derivatives if Φ (X,Y) with respect to the variables X and Y
N1 N2 N
respectively. So one of the axioms that we include is: if z =gz , then:
1 2
ad bc
Φ (j(z ),j(z ))j′(z )+Φ (j(z ),j(z ))j′(z ) − =0.
N1 1 2 1 N2 1 2 2 (cα(z )+d)2
1
(5) The axiom scheme (one L -statement for each N N): for all z ,z D,
ω1,ω ∈ 1 2 ∈
Φ (j(z ),j(z ))=0 = (gz =z ),
N 1 2 2 1
⇒
g∈G,det(_N(g)g)=N
so this axiom is a converse of axiom 4.
(6) The L -statement: let X G be set of non-scalar matrices. Then for every z D,
ω1,ω ⊆ ∈
(j(z)=0 j(z)=1728 j′(z)=0) = (gz =z).
∨ ∨ ⇒
g_∈X
Note that the values chosenare the same as those used in axiom 3, and correspondto the points
z where the expression k(j(z),j′(z),j′′(z),j′′′(z)) is not defined.
The name “j-field” and part of this definition come from work of Jonathan Kirby and Adam Harris
privately communicated to the author.
Note that from axiom 2 we get that α(D) Q = because the image of α is contained in K and for
∩ ∅
everyelementx Q there is g G suchthat gx= . Also,the axiomsfor α saythat D is embedded in
∈ ∈ ∞
K, and this allows us to identify D with α(D). Therefore we will avoid mentioning α in the future. For
the case of the modular j function, we take α:H+ H− C to be the inclusion.
∪ →
A SCHANUEL PROPERTY FOR j 5
Axioms 5 and 6 are not first-order expressible as we need a countable number of conjunctions. It
shouldnotbe toosurprisingthenthatthe setofrationalnumbersisdefinableinK. Toseethisnotefirst
that if a,b,c,d Q and r,s,t,u K are such that:
∈ ∈
ax+b rx+s
(3.2) =
cx+d tx+u
for all x D, then a/c=r/t. Now we can define the set V of tuples (r,s,t,u) K4 such that
∈ ∈
rx+s
(3.3) j =j(x)
(cid:18)tx+u(cid:19)
for allx D. This means,by axiom5 that there is a matrix g SL (Z), whose entries we name a,b,c,d
2
∈ ∈
in the usual way, that satisfy equation (3.2). After this set is defined, we know define the set of q K
∈
such that there exist r,s,t,u V satisfying r/t=q.
∈
Remark 1. The set of axioms we have given is not complete, we have only given the axioms that will be
needed for our transcendence results. The most natural way to define a complete set of axioms would
be to take the full theory of (C,H+ H−,j,j′,j′′,j′′′). Call this theory Th . From a model-theoretic
j
∪
perspective, we should only focus on models of Th , and maybe the name “j-fields” should be reserved
j
for these structures. Even if the theory Th is not the subject of this paper, we will point out a couple
j
of things.
(a) If (K,D) =Th , then any point of D whichis fixed by some g G is -definable (using terminology
j
| ∈ ∅
wewillintroduceshortly,this saysthatthe specialpointsare -definable),andsotheir imagesunder
∅
j are also -definable. As it turns out, these points are the only points at which the values of j
∅
algebraic over Q (this is the famous Schneider’s theorem, see [15]). Using some known values of j
one can prove that numbers such as √2, √3 and √5 are definable in K.
(b) Note that C (H+ H−) defines the real closed field R. So if (K,D) = Th , then K D is a real
j
\ ∪ | \
closed field. Note that it is not obvious how to associate to a given real closed field a corresponding
j function on its algebraic closure. See [11] to observe that the analytic construction of j shown in
the Preliminaries using genus one tori need not work in general. Also, given that real closed fields
are not categoricalnor ω-stable, it should not be expected that j-fields have these properties.
Because of this last point, and the fact that Q is definable, Th is not a very nice theory to work
j
with from a model-theory viewpoint. But we want to show that, even so, one can obtain interesting
number-theoretic results.
Definition. Let F be a subfield of K. We define GF to be the subgroup of GL (F) defined as GF :=
2
g GL (F):gD D . Note that GQ = G. F is called an active subfield of K if GF = GL (F). For
2 2
{ ∈ ⊆ }
example, in the case of K =C, every subfield of R is an active subfield of K.
A morphism of j-fields σ : (K ,D ) (K ,D ) is a field morphism σ : K K such that the
1 1 2 2 1 2
map σ := α−1 σ α : D D satisfi→es: for every f j,j′,j′′,j′′′ and for e→very z D we have
2 ◦ ◦ 1 1 → 2 ∈ { } ∈
σ(f (z))=f (σ(z)). Notethatfieldmorphismsrespectthe actionofthe groupG(F) onK,soweclearly
1 2
have σe(gz)=gσ(z) for every g G and z D.
Let (K,D) ebe a j-field and l∈et F be a ∈subfield of K. We say that F is a j-subfield of K if the pair
(F,De), whereeD = α−1(D F), is a j-field with the functions of (K,D) restricted to (F,D ). Note
F F F
∩
that if (K ) is a family of j-subfields of K, then K is a j-subfield of K.
i i∈I i∈I i
T
4. Geodesic Closures
In this section we study the pregeometry gcl which is defined by the action of GF on K, for every
F
subfield F of K.
Definition. Let A be a subset of K and F a subfield of K. We define the F-geodesic closure of A,
denoted gcl (A), as the set of x K such that there exist a A and g GF such that x =ga (which,
F ∈ ∈ ∈
savefor the exclusionofthe point atinfinity, is the unionof the GF-orbits ofpoints in A). When F =Q
we will simply write gcl(A).
It is straightforward to check that for every subfield F of K the operator gcl is a pregeometry of
F
trivial type. So given A,B K let dimg(A/B) be the dimension defined by the pregeometry gcl , i.e.
⊆ F F
the number ofdistinct orbits of elements in A thatdo notcontainelements of B. If B = then we write
simply dimm(A). ∅
F
6 SEBASTIANETEROVIĆ
It is also easy to see that if E is an active subfield of K, then dimg(E)=1. Note that if L F K
are subfields of K, then for every A K we have that dimg(A) diEmg(A). ⊆ ⊆
⊆ F ≤ L
LetE beasubfieldofK. Apointx K isE-special (orspecial over E)ifthereisanon-scalarg GE
∈ ∈
such that gx=x (when E =Q we will say simply that x is special).
4.1. GeodesicDisjointness. Thefollowingdefinitionofgeodesicdisjointnessisanalogoustothenotion
of linear disjointness (see e.g. Definition 3.1 of [3]).
Definition. Let E,F,L be subfields of K such that E F L. We say that F is geodesically disjoint
from L over E, denoted F gL, if for every tuple ℓ of⊆elem∩ents of L that is gcl -independent is also
⊥E E
gcl -independent. Alternatively, F gL if and only if for any tuple ℓ from L, dimg(ℓ)=dimg(ℓ).
F ⊥E F E
Lemma 4.1. Let E,L be subfields of K such that E L and E is active. Suppose that x is algebraically
⊆
independent over L and that GE =GE(x). Then:
6
(a) E(x) gL.
⊥E
(b) If ℓ L is special over E(x), then it is special over E.
∈
Proof. Let ℓ ,ℓ L (not necessarily distinct) and let g GE(x) GE be non-scalar. Suppose that
1 2
∈ ∈ \
ℓ =gℓ . First observe that as E(x) L=E, then either ℓ ,ℓ E or ℓ ,ℓ L E. As E is active, we
1 2 1 2 1 2
∩ ∈ ∈ \
assume that ℓ ,ℓ L E. Note that we can take the entries of g to be in E[x]. Write:
1 2
∈ \
a(x) b(x)
g = ,
(cid:18)c(x) d(x)(cid:19)
where a(x),b(x),c(x),d(x) E[x], they are not all constantpolynomials, and there is no common factor
∈
to all four polynomials. To set notation, write:
a(x)= a xi, b(x)= b xi, c(x)= c xi, d(x)= d xi;
i i i i i i i i
herewe are using the mulPti-index notation,Pso that i is a tuplPe ofnon-negativePintegers. We alsoconvene
that for all i, at least one of a , b , c , d is non-zero. From the equation ℓ =gℓ , we obtain:
i i i i 1 2
(c ℓ ℓ +d ℓ )xi = (a ℓ +β )xi.
i 1 2 i 1 i 2 i
Xi Xi
Given that a ,b ,c ,d E L and that x is algebraically independent over L, we deduce that the
i i i i
∈ ⊆
coefficient of xi is zero for every i. So we obtain that for all i:
a ℓ +b
i 2 i
ℓ = .
1
c ℓ +d
i 2 i
a b
Let g = i i . If det(g ) = 0 for some i, this would mean that the columns of g are linearly
i (cid:18)ci di(cid:19) i i
dependent over E, i.e. there exists λ E such that a =λ b and c =λ d . But this would mean that
i i i i i i i
∈
ℓ E.
1
∈
So det(g )=0 for all i. If every matrix g is scalar, then so would be g. Therefore there is some i for
i i
6
which g is non-scalar, and so ℓ and ℓ are gcl -dependent, and so, if ℓ = ℓ , then ℓ is special over
i 1 2 E 1 2 1
E. (cid:3)
Lemma 4.2. Let E,F,L be subfields of K such that E F L. Suppose F gL. Then for any tuple x
⊆ ∩ ⊥E
from K and any A L, we have:
⊆
dimg(x/L) dimg(x/L) dimg(x/A) dimg(x/A).
F − E ≤ F − E
Proof. Let ℓ L be a tuple such that dimg(x/ℓA) = dimg(x/L) and dimg(x/ℓA) = dimg(x/L). The
∈ F F E E
addition formula gives us two ways to calculate dimg(x,ℓ/A):
F
dimg(x,ℓ/A) = dimg(x/A)+dimg(ℓ/xA)
F F F
= dimg(ℓ/A)+dimg(x/ℓA).
F F
Therefore:
dimg(x/A) dimg(x/ℓA) = dimg(ℓ/A) dimg(ℓ/xA)
F − F F − F
= dimg(ℓ/A) dimg(ℓ/xA)
E − F
dimg(ℓ/A) dimg(ℓ/xA)
≥ E − E
= dimg(x/A) dimg(x/ℓA).
E − E
A SCHANUEL PROPERTY FOR j 7
(cid:3)
5. j-Derivations
In this section we introduce j-derivations and the pregeometry jcl they define. The exposition is
analogous to section 4 of [6].
Definition. Let K be a field and M a K-vector space. A map ∂ :K M is a called a derivation if it
→
satisfies for every a,b K:
∈
(1) ∂(a+b)=∂(a)+∂(b).
(2) ∂(ab)=a∂(b)+b∂(a).
Let (K,D) be a j-field. The map ∂ : K M is called a j-derivation if it is a derivation and it satisfies
→
for every z D:
∈
(3) ∂(j(z))=j′(z)∂(z), ∂(j′(z))=j′′(z)∂(z), ∂(j′′(z))=j′′′(z)∂(z).
For C K, let Der(K/C,M) denote the set of derivations ∂ : K M such that ∂(c) = 0 for every
⊆ →
c C. Let jDer(K/C,M) be the set of j-derivations ∂ : K M satisfying ∂(c) = 0 for every c C.
∈ → ∈
For convenience we write Der(K/C) := Der(K/C,K) and jDer(K/C) := jDer(K/C,K). Note that all
these spaces are K-vector spaces.
Let C be a subset of K. Define Ω(K/C) as the K-vector space generated by formal symbols of the
form dr, where r K, quotiented by the relations given by the axioms of derivations plus that for every
∈
c C,dc=0. Denotebyd:K Ω(K/C)themapr dr. Themapdiscalledtheuniversal derivation
∈ → 7→
on C. Let Ξ(K/C) be obtained from Ω(K/C) by taking the quotient with the axioms for j-derivations.
This induces a map d :K Ξ(K/C) which is called the universal j-derivation.
j
→
Proposition 5.1. Let C K. For every j-derivation ∂ : K M which vanishes on C, there exists a
⊆ →
unique K-linear map ∂∗ :Ξ(K/C) M such that: ∂∗ ∂ =∂.
j
→ ◦
Proof. For every r K, set ∂∗(d r)=∂(r) and then extend linearly. (cid:3)
j
∈
Clearly, a similar propositionexists characterisingthe universalproperty of Ω(K/C). These universal
properties in fact show that Der(K/C) and jDer(K/C) are the dual spaces of Ω(K/C) and Ξ(K/C)
respectively.
Definition. Let C K, and a K. We say that a belongs to the j-closure of C, denoted a jcl(C) if
⊆ ∈ ∈
for every ∂ jDer(K/C) we have that ∂(a)=0. That is:
∈
jcl(A)= ker∂.
∂∈jD\er(K/A)
In particular this means that for any A K and any ∂ jDer(K/A) we have that ker∂ is jcl-closed.
⊆ ∈
For any C K, F = jcl(C) is a j-subfield of K. As every j-derivation is an additive homomorphism,
⊆
jcl(C)isagroupunderaddition. Thesecondaxiomofj-derivationsshowthatitisclosedundermultipli-
cation. Finally, as 1 is in the kernel of any j-derivation, then the multiplicative inverse of every nonzero
element of F is in F. So F is a field. The other axioms of j-derivations show that we can restrict j to
F and still obtain a j-field. For this, first define D =D F. Thus we can define α :D F as the
F F F
∩ →
restriction of α to D . Let z D . Let ∂ jDer(K/C), then as ∂(j(z)) = j′(z)∂(α(z)) we get that
F F
∈ ∈
∂(j(z))=0, and so j(z) F. Similarly j′(z),j′′(z) F. The fact that j′′′(z) F comes from axiom (3)
∈ ∈ ∈
ofj-fields,andthe factthat F is relativelyalgebraicallyclosedinK. The finalthing to checkis thatD
F
is a G-set. But this follows because for every g G, α (gz) is algebraic over F (because α (z) F),
F F
∈ ∈
and so ∂(α (gz))=0. Therefore (F,D ) is a j-subfield of K.
F F
Remark 2. Note that a jcl(C) if and only if d a = 0 in Ξ(K/C). Clearly if d a = 0, then a jcl(C).
j j
∈ ∈
Conversely suppose that d a = 0. Assuming Ξ(K/C) = 0, we can choose a linear transformation
j
6 6
η :Ξ(K/C) K suchthatη(d a)=0. Itis a straightforwardexerciseto showthat η d jDer(K/C).
j j
→ 6 ◦ ∈
Therefore,thereexists∂ jDer(K/C)suchthat∂(a)=0. FromthiswecanconcludethatjDer(K/C)=
∈ 6
jDer(K/jcl(C)).
Recall the following standard results about derivations.
Lemma 5.2 (see [7]). Let K be a field and ∂ :K K be a derivation.
→
8 SEBASTIANETEROVIĆ
(1) Let C K and let a K be algebraic over Q(C). Then every derivation which vanishes on C,
⊆ ∈
also vanishes on a.
(2) Let X be transcendental over K and let a K(X). Then ∂ can be extended to a derivation
∈
∂′ :K(X) K(X) such that ∂′(X)=a.
→
(3) If L is a separable algebraic extension of K, then ∂ extends to a derivation on L.
Lemma 5.3 (see Lemma 6.7 of [9]). Let K K be a field extension. Then dim Ω(K /K ) =
1 ⊆ 2 K2 2 1
t.d.(K /K ).
2 1
Lemma5.4 (seeTheorem3of[2]). LetK beafieldand∆asetofderivations on K. Letx ,...,x K
1 n
∈
and set r = rk(∂x ) . Then there is a set of derivations ∆ and ∂′,...,∂′ ∆ such that
i ∂∈∆,i=1,...,n 1 n ∈
∂′(x )=δ , the Kronecker delta, for every ∂ ∆ ∂′,...,∂′ we have that ∂(x )=0 for i=1,...,n,
i k ik ∈ \{ 1 n} e i e
and the elements of ∆ are K-linear combinations of the elements of ∆. Furthermore ker∂ =
e ∂∈∆
∂∈∆e ker∂. In particuelar rk(∂xi)∂∈∆e,i=1...,n =r. T
T
Proof. Assumethat∂ ,...,∂ ∆andx ,...,x aresuchthat: det(∂ x ) =0. LetA=(a )=
1 r 1 r i k i,k=1,...,r ik
(∂ x )−1 and set ∂′ = ∈r a ∂ Der(F/C). In this way ∂′(x )=δ , the6Kroneckerdelta.
i k i,k=1,...,r i k=1 ik k ∈ i k ik
For each∂ ∆there existPunique b (∂) F suchthat: ∂(x )= r b (∂)∂ (y )for k =1,...,r. Set
∈ i ∈ k i=1 i i k
∂ =∂ r b (∂)∂ and ∆= ∂ :∂ ∆ ∂′,...,∂′ . P
− i=1 i i n ∈ o∪{ 1 r}
e Let CP= ker∂. Leet x eK be such that ∂(x) = 0 for every ∂ ∆. In particular ∂′(x) = 0, for
∂∈∆ ∈ ∈ i
i=1,...,r.TGiven that A is invertible, we get that ∂i(x)=0 for i=1,..e.,r. Therefore, for each ∂ ∈∆
we have that 0=∂(x)=∂(x) r b (∂)∂ (x)=∂(x), which means that x C.
− i=1 i i ∈
Conversely, it is immediate froPm definition that if x C, then ∂(x)=0 for every ∂ ∆. (cid:3)
e ∈ ∈
Definition. Let (K,D) be a j-field and let τ ,...,τ K be jcl-independent. A correspeonding system
1 m
∈
of j-derivations for τ isasetofj-derivations∂ ,...,∂ jDer(K)suchthat∂ (τ )=δ ,theKronecker
1 m i k ik
∈
delta.
So, whatLemma 5.4shows (if one restricts the proofso that it only uses j-derivations)is that for any
jcl-independent set, there is a correspondingsystem of j-derivations. Now we will focus on properties of
j-derivations. The following Lemma is straightforward.
Lemma 5.5. Given subsets B,C K we have that:
⊆
(a) C jcl(C).
⊆
(b) B C = jcl(B) jcl(C).
⊆ ⇒ ⊆
(c) jcl(jcl(C))=jcl(C).
Lemma 5.6 (Finite character of jcl). Let (K,D) be a j-field, C K. If a jcl(C), then there is a
⊆ ∈
finite set C C such that a jcl(C ). Furthermore, there is a finitely generated j-subfield K of K
0 0 0
⊆ ∈
such that C K and a jcl (C ).
0 ⊆ 0 ∈ K0 0
Proof. Let be the language that consists purely of constant symbols, one for each element of Ξ(K/ ).
L ∅
Let T be the -theory that says that these constant symbols satisfy the axioms of K-vector spaces,
L
the axioms that say that d is a j-derivations, and also d c = 0 for each c C. Then Ξ(K/C) = T
j j
∈ |
and Ξ(K/C) = d a = 0 by Remark 2. By the universal property of Ξ(K/C), every other j-derivation
j
|
∂ :K M whichvanishes overC alsosatisfies ∂(a)=0. Also, everysuchM is a model ofT. Therefore
→
T = d a = 0. By the Completeness Theorem T d a = 0. There are only finitely many elements of C
j j
| ⊢
usedintheproofofd a=0. LetT bethesetofformulasusedintheproofofd a=0(thus,T isfinite).
j 0 j 0
Let C consist of the elements c C such that the formula d c=0 T . Let K be the j-subfield of K
0 j 0 0
∈ ∈
generated by all the x K which appear in some axiom of T . Note that C K . Then d a = 0 in
0 0 0 j
Ξ(K /C ) and in Ξ(K/∈C ). Therefore a jcl (C ) and a jcl (C ). ⊆ (cid:3)
0 0 0 ∈ K0 0 ∈ K 0
Lemma 5.7 (Exchange Principle for jcl). Let C K, a,b K. If a jcl(Cb) and a / jcl(C), then
⊆ ∈ ∈ ∈
b jcl(Ca).
∈
Proof. Suppose a jcl(Cb) and b / jcl(Ca). Then there is ∂ jDer(K/C) such that ∂(a) = 0 and
∈ ∈ ∈
∂(b)=1. Let∂′ jDer(K/C)andlet∂′′ =∂′ (∂′(b))∂. Then∂′′(a)=∂′(a) (∂′(b))∂(a)=∂′(a). Also
∈ − −
∂′′(b)=∂′(b) (∂′(b))∂(b)=0. So ∂′′ jDer(K/Cb), which means that ∂′′(a)=0, therefore ∂′(a)=0.
Therefore a −jcl(C). ∈ (cid:3)
∈
A SCHANUEL PROPERTY FOR j 9
Combining these lemmas we get that jcl defines a pregeometry on any j-field.
Definition. C K is said to be jcl-closed if C = jcl(C). C is said to be j-independent if it is
⊆
independentwithrespecttojcl. Inparticular,ifAisjcl-independentandhasonlyoneelementA= τ ,
{ }
then we will say τ is j-generic. Furthermore, for A,C K, let dimj(A/C) denote the dimension of A
⊆
over C with respect to jcl.
The next proposition should enlighten the reader as to the name chosen for the operator jcl.
Proposition 5.8. Let C K be jcl-closed, and let z D.
⊆ ∈
(a) α(z) C = j(z),j′(z),j′′(z) C
∈ ⇒ { }⊆
(b) Suppose j′′′(z)=0. If j(t)(z) C for some t 0,1,2 , then α(z) C.
6 ∈ ∈{ } ∈
Proof. (a) Let ∂ jDer(K/C). If ∂(α(z)) = 0, then ∂(j(t)(z)) = 0, for t 0,1,2 , by axioms of
∈ ∈ { }
j-derivations.
(b) Let ∂ jDer(K/C). Using the equations ∂(j(t)(z)) = j(t+1)(z)∂(α(z)), for t 0,1,2 , we get
∈ ∈ { }
∂(α(z))=0.
(cid:3)
Remark 3. The specialpoints overQ arein jcl( ). This is because the equation gx=x is either a linear
∅
or quadratic equation over Q, so special points are of degree 2 over Q.
Proposition 5.9. Let z ,...,z D be such that z ,...,z are jcl-independent over C K. Then we
1 n 1 n
have that dimg(z ,...,z /C)=n.∈ ⊆
1 n
Proof. Clearly dimg(z ,...,z /C) n. Suppose that dimg(z ,...,z /C) < n. Then, without loss of
1 n 1 n
≤
generality,wemayassumethatz gcl(C,z ,...,z ). This meansthatthereis x C z ,...,z and
1 2 n 2 n
∈ ∈ ∪{ }
g G such that z = gx. Every j-derivation which vanishes on x vanishes also on z . This means that
1 1
z ∈ jcl(x), which contradicts that z ,...,z are jcl-independent over C. (cid:3)
1 1 n
∈
Lemma 5.10. Let (K,D) be a j-field and C K. Then dimj(K/C) dim Ξ(K/C).
K
⊆ ≤
Proof. Suppose t ,...,t are jcl-independent over C. Then, as in Lemma 5.4, there are j-derivations
1 n
∂ : K K such that ∂ (t ) = δ , the Kronecker delta. Consider the universal j-derivation d : K
i i k ik j
→ →
Ξ(K/C) and for each i choose χ : Ξ(K/C) K such that ∂ = χ d . Suppose that a ,...,a K
i i i j 1 n
→ ◦ ∈
satisfy:
n
a d t =0.
i j i
Xi=1
Then, applying χ to this equation we get a ∂ t = 0, that is to say a = 0. Therefore d t ,...,d t
k i k i k j 1 j n
are K-linearly independent. P (cid:3)
To summarise the results about dimensions of universal derivations, we write:
dimj(K/C) dim Ξ(K/C)=dim Ξ(K/jcl(C)) dim Ω(K/jcl(C))=t.d.(K/jcl(C)).
K K K
≤ ≤
5.1. j-derivations on C. In this section we prove that there are non-trivial j-derivations on C. It is
well known that j has a standard fundamental domain:
1
:= z =x+iy H+ : x , z 1 .
F (cid:26) ∈ | |≤ 2 | |≥ (cid:27)
For every z H+ there exists g SL (Z) such that gz . Any set of the form g , with g SL (Z)
2 2
∈ ∈ ∈ F F ∈
is called a fundamental domain of j. Now note that the differential equation of j (2.2) is defined when
j′(z)=0 and j(z)(j(z) 1728)=0. When working on C, we know what are the values of z that do
6 − 6 ∈F
this:
if j′(z)=0, then z =i or z =ρ, where ρ=e2πi/3,
•
if j(z)=0, then z =ρ,
•
if j(z)=1728, then z =i.
•
Proposition 5.11. Let ∂ be a j-derivation on C. Let z H+ H not be in the SL (Z)-orbits of i, i,
2
∈ ∪ −
ρ and ρ. Then for every n N we get that:
∈
∂(j(n)(z))=j(n+1)∂(z).
10 SEBASTIANETEROVIĆ
Proof. To prove the result for n 3, we express equation (2.2) in the following form:
≥
3(j′′)2 j2 1968j+2654208
(5.1) j′′′ = − (j′)3.
2j′ − 2j2(j 1728)2
−
Now we do two things to this equation: on one hand derive it (as holomorphic functions), on the other,
apply∂ toit. Comparebothresultstogetthat∂(j′′′)=j(4)(z)∂(z). Byinduction,weobtainthe desired
result. (cid:3)
Given that i and ρ are algebraic over Q, we get the following corollary.
Corollary 5.12. For every n N, if j(n)(z)=0, then z jcl( ).
∈ ∈ ∅
Proof. Suppose not,andlet∂ be aj-derivationsuchthat∂(z)=0. The orderofazeroofa holomorphic
6
function is finite, so after applying ∂ to j(n)(z) enough consecutive times, we will get ∂(z) = 0, a
contradiction. (cid:3)
Now we look for non-trivial j-derivations. The key idea here is that, to get j-derivations, we should
look for derivations on R that respect the real and imaginary parts of j on its fundamental domain. Let
us set the following notation. If F : U C is a holomorphic function, where U C, then the real and
→ ⊆
imaginary parts of F, say F and F respectively, are the real analytic functions with domain:
1 2
U := (x,y) R2 :x+iy U
R
∈ ∈
(cid:8) (cid:9)
satisfying F(x+iy)=F (x,y)+iF (x,y), for (x,y) U .
1 2 R
∈
Given two functions f ,f : R R, define the function [f : f ] : C C as [f : f ](x+iy) :=
1 2 1 2 1 2
→ →
(f (x) f (y))+i(f (y)+f (x)) (for x,y R).
1 2 1 2
− ∈
Definition. Let ∂ be a derivation on C and F : U C be a holomorphic function. We say that ∂
→
respects F at a point z U if:
∈
dF
∂(F(z))= (z)∂(z).
dz
If F is the real or imaginary part of F and ∂′ is a derivation on R, then we say that ∂′ respects F at
0 0
z =x+iy U if:
∈
∂′(F (x,y))=∂ (F (x,y))∂′(x)+∂ (F (x,y))∂′(y),
0 1 0 2 0
where ∂ ,∂ are the partial derivatives of F with respect to the first and second variables fo F respec-
1 2 0 0
tively.
The next lemma tells us that to obtain j-derivations on C we can use j-derivations on R.
Lemma 5.13 (see Lemma 4.2 of [19]). If ∂ ,∂ are derivations on R, then [∂ : ∂ ] is a derivation on
1 2 1 2
C. Further, if F is a holomorphic function with real and imaginary parts F ,F and domain U C, and
1 2
⊆
if ∂ ,∂ respect F and F at a point (x,y) U , then [∂ :∂ ] respects F at the point x+iy.
1 2 1 2 R 1 1
∈
Let F : U C be a holomorphic function. The Schwarz reflection of F is the holomorphic func-
→
tion FSR : U′ C given by FSR(z) := F(z), where U′ = z :z U and the bar denotes complex
→ { ∈ }
conjugation. The following theorem determines the kind of j-derivations we could hope to find on C.
Theorem 5.14 (see Theorem4.3 of[19]). Let be any collection of holomorphic functions closed under
C
Schwarz reflection and holomorphic derivation. Then the elements of Der ( ) are precisely the maps of
C
C
the form [λ:µ]:C C, for λ,µ Der ( ).
R real
→ ∈ C
LetR be the expansionofR with the realandimaginaryparts ofj,j′ andj′′, withallthese functions
restricted to . It is noted in [11] that the function j : F C is definable in R . So the structure
an,exp
F →
R is o-minimal. The next lemma is a basic result from complex analysis.
Lemma 5.15. Let f : U C be a holomorphic function. Write f(x+iy) = u(x,y)+iv(x,y), for
→
(x,y) U . Then:
R
∈
df df
e =u , m =v ,
x x
R (cid:18)dz(cid:19) I (cid:18)dz(cid:19)
where u and v denote the partial derivatives of u and v with respect to the first variable.
x x
Proposition 5.16. The structure R has non-trivial derivations that respect the real and imaginary parts
of j, j′ and j′′.