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THEOREMS ON UNLIKELY INTERSECTIONS BY COUNTINGPOINTS IN DEFINABLE SETS

THOMAS SCANLON

1. COMMENTS ON THESE NOTES

This document consists of my notes prepared in advance of a series oflectures at Luminy as part of the program on the Zilber-Pink conjecture.Since I did not meet the deadline imposed by the institute administrators,many parts of this paper are incomplete. Let me warn the reader that thereare certainly errors in these notes, references are missing and they are notorganized as I would wish. I plan to distribute a better version just beforethe lectures begin.

I have delivered a couple of general audience lectures about this topicand have prepared accompanying expository papers [21, 20]. The readerdesiring a smoother presentation may prefer to read those notes and willlikely notice that some of the present text was lifted from those sources.While I have made an effort to present some of the arguments in greaterdetail in these notes than in the earlier notes, the proofs are far from com-plete. The original papers on which this material is based are clearly writ-ten. I recommend that the reader study those papers directly.

2. AN OUTLINE OF THE PILA-ZANNIER STRATEGY

With this lecture we shall describe the key steps in the Pila-Zannier strat-egy leaving most of the key details to later lectures. This strategy was firstsuccessfully implemented by Pila and Zannier in [16] with their reproof ofthe Manin-Mumford conjecture (Raynaud’s theorem [19]). Shortly there-after, Masser and Zannier employed these ideas to give a proof of the firstnon-trivial instance of the Zilber-Pink conjecture for which the special va-rieties have positive dimension. Subsequently, various authors have ex-tended these methods to prove other cases of the Zilber-Pink conjecture,most notably, Pila with his unconditional proof of the Andre-Oort conjec-ture for products of modular curves [15].

Let us consider two abstract forms of the Zilber-Pink conjecture. Thefirst theorem template might be considered as an abstract version of theManin-Mumford or Andre-Oort conjecture.

Partially supported by NSF grants FRG DMS-0854998 and DMS-1001550.1

2 THOMAS SCANLON

Theorem Template 2.1. Let X be a special variety over a field K and Σ ⊆ X(K)a set of special points on X, then the Zariski closure of Σ is a finite union ofspecial subvarieties.

Of course, Theorem Template 2.1 has no content unless the terms “spe-cial point” and “special variety” are given some geometric meaning. In thecase that X is a semiabelian variety over K = C, if one were to read “torsionpoint” for “special point” and “translate of a group subvariety by a torsionpoint” for “special variety,” then this instance of Theorem Template 2.1 isprecisely the Manin-Mumford conjecture. If we take X to be a Shimuravariety over K = C, then, as written, Theorem Template 2.1 is the Andre-Oort conjecture where the terms “special point” and “special variety” aredefined relative to Deligne’s theory of Shimura data.

An abstract form of the Zilber-Pink conjecture takes the following form.

Theorem Template 2.2. Let X be a special variety over a field K and Y ⊆ X anarbitrary subvariety. If Y is not a special subvariety of X, then the following set isnot Zariski dense in Y. ⋃

Z ⊆ XZ a special subvariety

dim(Z) + dim(Y) < dim(X)

Z(K) ∩Y(K)

As with Theorem Template 2.1, Theorem Template 2.2 asserts somethingnontrivial only when the notions of special variety and special subvarietyare made precise. Visibly, Theorem Template 2.2 (for a given X, Y and inter-pretation of the word “special”) implies Theorem Template 2.1. While thisformal implication is undeniable, there is a sense in which these theoremtemplates have qualitatively different characters. Theorem Template 2.1describes the possible algebraic relations on the set of special points of Xwhile Theorem Template 2.2 is more geometric in that it predicts the pos-sible interactions between special algebraic relations and general algebraicrelations. Paradoxically, methods from o-minimality are better suited toinstances of Theorem Template 2.2 than they are to Theorem Template 2.1in that in many cases of interest questions the class of special subvarietiesof a given variety may be realized as the restriction of a definable fam-ily of definable sets. The issues implicated by Theorem Template 2.2 arethereby transformed into questions about definable sets. Admittedly, thiscomment is too vague to be meaningful and, from our earlier observation,facially false, but I hope that by the end of these lectures the sense in whichI intend it has been made clear.

Let us sketch the Pila-Zannier method applied to Theorem Template 2.1.So that the reader may follow the proof of a true theorem we shall in-stantiate Theorem Template with a particularly simple case of the Manin-Mumford conjecture, namely, Mann’s theorem about algebraic relation onroots of unity [9].

COUNTING POINTS IN DEFINABLE SETS 3

Theorem 2.3. Let g ≥ 1 be a positive integer and Y ⊆ Ggm a subvariety of the gth

Cartesian power of the multiplicative group over the complex numbers. Then theset

{(ζ1, . . . , ζg) ∈ Y(C) : ζi is a root of unity for each i ≤ g}is a finite union of cosets of subgroups of (C×)g.

Remark 2.4. Theorem 2.3 is an instance of Theorem Template 2.1 in whichwe take the ambient special variety X to be G

gm, the special points to be

g-tuples of roots of unity, Σ to be the displayed set in the statement ofTheorem 2.3 and the special subvarieties to be the components of groupsubvarieties of G

gm.

Remark 2.5. There are several non-trivial difficulties with implementing thePila-Zannier strategy in general which disappear in this special case we areconsidering now. In the later lectures we shall go into detail about theseother theorems.

The first step of this method is to find some analytic covering map π :X → X(C). For this to be a useful move, we shall require some strongerproperties of π and X than merely that π be analytic. Returning to thecase of Mann’s theorem, X = Cg and the map π : Cg → G

gm(C) is simply

(z1, . . . , zg) 7→ (e2πiz1 , . . . , e2πizn) where X = Cg.One requirement for the map π : X → X(C) to be useful is that the

inverse image of the special points in X(C) under π, what we shall call pre-special points, should also be “special points” in X. Again thinking of thecase of Mann’s theorem, the special points on X(C) are g-tuples of roots ofunity and the set of pre-special points is precisely Qg.

Assuming that the theorem is false, replacing Y by a component of somenonspecial component of the Zariski closed of the set of special points on Y,we may assume that Y is not a special subvariety but that the set of specialpoints on Y is Zariski dense in Y. Let Y := π−1Y(C). Then Y is a (notnecessarily irreducible) analytic subvariety of X which contains “many”pre-special points. Returning to Mann’s theorem, Y contains many rationalpoints. That is, we have reduced the problem of describing the specialpoints on Y(C) to that of describing the pre-special points on the analyticvariety Y.

At this point in the sketch, the reader would be justified in rejecting thisstrategy out of hand as there is almost nothing that one can say about ra-tional points on analytic sets in general. On the one hand, for any setS ⊆ Zg we can find an entire analytic function f (x1, . . . , xg) for whichf (z1, . . . , zg) = 0 just in case (z1, . . . , zg) ∈ S. Thus, there is no restric-tion at all on a subset of Zg to be the rational points on an analytic variety.On the other hand, even if we insist that Z ⊆ Cg is a particularly well be-haved analytic set, then it may be very hard to describe Z ∩Qg. Indeed, ifwe ask that Z is the set of complex points of an algebraic variety, then prob-lem of determining Z ∩Qg is the central task of arithmetic geometry and

4 THOMAS SCANLON

is (conjecturally) an intractable problem. So, it would seem that we haveconverted the difficult problem of determining the set of special points onY to the impossible task of classifying the possible sets of pre-special pointson the analytic variety Y.

For our argument to succeed, we must steer clear of these extremes of acompletely general analytic variety and algebraic varieties. We avoid thepathologies of general analytic varieties by requiring (a suitable restrictionof) π to be definable in an o-minimal expansion of the real numbers. We skirtthe problems in arithmetic geometry by showing that for purposes of thisargument, we may simply ignore the rational points contained in algebraicsubvarieties.

Let us delay a detailed discussion of definability and take the notion ofa set definable in an o-minimal expansion of the real numbers as a blackbox for the moment, though at least to keep this sketch honest we shouldmake the suitable restriction somewhat more explicit. Even in this spe-cial case of Mann’s theorem where the function π is simply given by thecomplex exponential function, one must limit the domain of π in order tohave any hope of its logical theory being reasonable. Indeed, using thecomplex exponential function itself, since one may define the integers as{z ∈ C : e2πiz = 1}, one may encode all of arithmetic in the theory ofthe complex field given together with the exponential function. As followsfrom Godel’s Incompleteness theorem, the resulting theory is wild. On theother hand, for the purposes of recognizing roots of unity as special val-ues of the exponential function, since the exponential function is periodic,it is not necessary to consider the action of the exponential function every-where. In the case of Mann’s theorem, we could take X := {(z1, . . . , zg) ∈Cg : 0 ≤ Re(zi) < 1 for each i ≤ g}. With such a restriction the functionπ : X→ X(C) = (C×)g is definable in the o-minimal structure Ran.

Remark 2.6. The astute reader has observed that in the case of Mann’s the-orem, one could work with an even smaller domain. That is, if ea is root ofunity, then a is a rational multiple of πi. Thus, if we were to take X := [0, 1]g

and ν : X→ X(C) defined by (x1, . . . , xg) 7→ (exp(2πix1), . . . , exp(2πixg)),then it already the case that every special point in (C×)g is in the image ofν. Thus, we could analyze the set of special points on the variety Y ⊆ G

gm

as the image under ν of the set of rational points on ν−1Y(C). While thismove would simplify the argument for Mann’s theorem, it is somewhatspurious as one cannot perform such a reduction for most of the other cases(eg Andre-Oort or Manin-Mumford) to which this method applies.

For purposes of this argument, the most important consequence of Ybeing definable in a o-minimal structure on the real numbers is a countingtheorem of Pila and Wilkie asserting that if one avoids its algebraic part,there are few rational points on Y. Before we make this result precise, letus return to the problem at hand to see how such a theorem may be used.Our algebraic variety Y is not special, but it contains a Zariski dense set of


special points. We shall show that the image under π of the algebraic part ofY is contained in a proper subvariety of Y. We then conclude that there aremany pre-special points in Y outside of its algebraic part where “many”means that the image of this set under π is Zariski dense in Y. The Pila-Wilkie counting theorem is quantitative and on the face of it, there could bea set of points with Zariski dense image in Y but whose distribution is sosparse that one would not observe a contradiction. To reach a contradiction,we must employ a Galois theoretic argument to show that in this case, suchsets must be fairly large.

Let us flesh out this argument beginning with a precise statement of thecounting theorem. For this we need to say what we mean by the algebraicpart of a set.

Definition 2.7. We say that a subset of Rn is semi-algebraic if it is a finiteboolean combination of sets of the form {(x1, . . . , xn) ∈ Rn : f (x1, . . . , xn) ≥0} where f is a polynomial over the real numbers. We say that a subsetof Cn is semi-algebraic if its image in R2n under the map (z1, . . . , zn) 7→(Re(z1), Im(z1), . . . , Re(zn), Im(zn)) is semi-algebraic. A function is semi-algebraic if its graph is. Given a set Y ⊆ Cn we define the algebraic partof Y, written Yalg, to be the union of the ranges of all nonconstant semi-algebraic functions γ : (0, 1) → Y. We define the transcendental part of Y,written Ytr, to be Y rYalg.

Remark 2.8. We shall go into much more detail about the construction of thealgebraic part of a set in the third lecture. For now, let us simply observethat it follows from a curve selection theorem in semi-algebraic geometrythat if Y is itself semi-algebraic, connected and infinite, then it is equal toits own algebraic part.

Let us say now how we shall count the rational points in a definableset. Recall that the (multiplicative) height of a rational numbers is definedby H(0) = 0 and H( a

b ) := max{|a|, |b|} provided that gcd(a, b) = 1. Weextend H to n-tuples by H(x1, . . . , xn) := max{H(xi) : 1 ≤ i ≤ n}. Fora set X ⊆ Rn and t ∈ R+, we define X(t) := {a ∈ X : H(a) ≤ t}. In itssimplest form, the Pila-Wilkie counting theorem asserts that if X ⊆ Rn is o-minimally definable, then there are subexponentially many rational pointson the transcendental part of X.

Theorem 2.9 (Pila-Wilkie). Let X ⊆ Rn be o-minimally definable and ε > 0.Then there is a constant C = C(X, ε) so that for all numbers t ≥ 1 we have#Xtr(t) ≤ Ctε

Returning to our sketch of the strategy to prove Theorem Template 2.1,to use Theorem 2.9 we should first determine Yalg, relating this set to thespecial subvarieties of Y. For Mann’s theorem, this is achieved by applyinga function field version of the Schanuel conjecture due to Ax.

6 THOMAS SCANLON

Theorem 2.10. Suppose that γ1(t), . . . , γn(t) ∈ tC[[t]] are complex power serieswith no constant terms which are linearly independent over Q. Then the transcen-dence degree over C of the field C(γ1, . . . , γn, exp(γ1), . . . , exp(γn)) is at leastn + 1.

We use Theorem 2.10 to show that the image of Yalg under π is the Uenolocus of Y:

Ueno(Y) :=⋃

aH ⊆ XH ≤ G

gm

dim(H) > 0

aH

Remark 2.11. While it is not immediately obvious from the definition, it isnot hard to show that the Ueno locus is actually a (not necessarily irre-ducible) subvariety of Y and that Y = Ueno(Y) if and only if the stabilizerof Y is infinite.

Of course, if π(ξ) ∈ Ueno(Y), then ξ ∈ Yalg as there is a line with ra-tional slope passing through ξ and contained in Y. Let us show now thatevery semialgebraic curve on Y lies over the Ueno locus. Suppose now thatγ : (0, 1) → Y is a nonconstant semialgebraic function. Applying someminor reductions (decomposing (0, 1) into finitely many intervals and pos-sibly precomposing with a monomial), we may assume that γ is real an-alytic on an interval properly containing (0, 1). Choosing coordinates onCg around γ(0), we may write γ in coordinates as (γ1(t), . . . , γg(t)) whereγi(0) = 0 for each i ≤ g. Since the image of γ is contained in Y, applyingπ : X→ G

gm(C), we see that (exp(2πiγ1(t)), . . . , exp(2πiγg(t)) ∈ Y(C) for

all t ∈ (0, 1). Thus, the functions exp(2πiγiγ1(t)), . . . , exp(2πiγg(t)) arealgebraically dependent over C. As γ is semialgebraic, the transcendencedegree over C of C(γ1(t), . . . , γg(t)) = 1. Thus,

tr. degC C(γ1(t), . . . , γg(t), exp(2πiγ1(t)), . . . , exp(2πiγg(t))) ≤ g

so that by Theorem 2.10 the functions γ1, . . . , γg must be linearly depen-dent over Q, but the image of such a translate of a vector space definedby a the vanishing of a linear form over Q under π is simply a translateof an algebraic subgroup variety of G

gm. Let us call this subgroup H and

π(γ(0)) =: a. Passing to a connected component of H if need be, we knowthat H ∼= Gt

m for some t < g. Via this isomorphism composed with a trans-lation, we may regard Y ∩ aH as a subvariety of Gt

m and the curve γ as asemialgebraic curve on the preimage of this variety. Hence, by induction,the image of γ is contained in the Ueno locus of aH ∩Y, which is containedin the Ueno locus of Y.

Remark 2.12. In the general case, various ideas may come to bear on thispart of the problem of computing Yalg. As with Mann’s theorem, analoguesof Ax’s theorem are known for the exponential maps associated to abelian


varieties and an argument like the one sketched above may be used to showthat Yalg is the preimage of the Ueno locus of Y. Ax’s proof is differentialalgebraic in nature and one might expect that suitable analogues for thecovering maps appearing in the theory of Shimura varieties, for example,could be proven through a consideration of the differential equations theysatisfy. However, to date the differential algebraic techniques have not ledto complete proofs. In Pila’s proof of the Andre-Oort conjecture for prod-ucts of modular curves, Yalg is characterized through another applicationof the counting theorem. With the Masser-Zannier theorem on simultane-ous torsion, results on transcendence of theta functions are used.

Quotienting by the stabilizer of Y if need be, we reduce the problem toshowing that there are only finitely many special points on the complementto Ueno(Y) in Y. Equivalently, since we know that Yalg is the preimage ofthe Ueno locus of Y, we must show that there can be only finitely manyrational points in Ytr.

Let us observe that since Y contains a Zariski dense set of special points,each of which is algebraic, Y itself is defined over some number field, L.Moreover, it is clear that Ueno(Y) is preserved by the action of the Gal(Qalg/L).Thus, if ξ ∈ Ytr ∩Qg and σ ∈ Gal(Qalg/L), there some point ξσ ∈ Ytr ∩Qg

for which σ(π(ξ)) = π(ξσ)). In the case, it is very easy to describe thepoint ξσ. Indeed, we may write ξ = ( a1

b1, . . . , ag

bg) with 0 ≤ ai < bi, ai, bi ∈ Z,

(ai, bi) = 1 unless ai = 0 and bi = 1. Then σ(ξ) = (a′1b1

, . . . ,a′gbg) with a′i < bi

and coprime to bi unless ai = a′i = 0. As we know, if n > 2 and ζn is aprimitive nth root of unity, then [Q(ζn) : Q] = ϕ(n) where ϕ is Euler’s to-tient function. Moreover, it is easy matter to show that for n big enough,ϕ(n) > n/ log(n) > n

12 .

Let C = C(Y, 13 ) be the constant of Theorem 2.9. It follows that Ytr ∩

Qg ⊆ Y((C[L : Q])6) as if there were some point ξ ∈ Ytr ∩Qg with H(ξ) =n > (C[K : Q])6, then from the above calculation of the size of the Galoisorbit of π(ξ), we see that there would be at least 1

[K:Q]n

12 points in Ytr(n)

while the counting theorem limits this set to Ct13 points which contradicts

the hypothesis that n > (C[K : Q])6.This completes the sketch of the Pila-Zannier strategy implemented in

the case of Mann’s theorem. �For the most part the instances of Theorem Template 2.2 proven via this

strategy follow the general form of the argument we sketched for Mann’stheorem. This is true even for the Masser-Zannier theorem on simultaneoustorsion which is not an instance of the weaker Theorem Template 2.1. Therecent work of Habegger and Pila on anomalous intersections in Shimuravarieties is the first case where the natural extension of these methods tothe more general Zilber-Pink conjecture has been applied successfully.

8 THOMAS SCANLON

With our proof sketch, not only did we leave the notion of o-minimaldefinability undefined, but we did not even so much as indicate why thePila-Wilkie counting theorem might be true. With the following lectures,we shall fill this gap. With the theory of o-minimality and its connectionto arithmetic firmly established, we shall return to the more sophisticatedapplications to problems in diophantine geometry.

3. O-MINIMAL GEOMETRY

O-minimality is a logical condition isolated by van den Dries [27] fromwhich the theory of semi-algebraic geometry may be developed axiomat-ically, and ultimately, generalized. In order to express the definition of o-minimality we require some terminology from mathematical logic and Iwould argue that to appreciate the strength of o-minimality one should ap-proach the subject with a sensibility informed by logic. In earlier surveys ofthis topic, I made a point of suppressing the logical apparatus, but because Iwish to explain in some detail how the Pila-Wilkie counting theorem reallyworks, logic will take the foreground in these notes.

First, we need to make sense of definability. There are at least two stan-dard ways to say what we mean by a structure in the sense of first-orderlogic. One can define a structure to be a nonempty set equipped with sys-tem of subsets of its Cartesian powers closed under natural operations. Inso doing, it becomes unnecessary to discuss syntax.

Definition 3.1. A structure M consists of a nonempty set M given togetherwith a collection Dn of subsets of Mn for each n ∈ Z+. A set X ∈ Dn is saidto be a definable subset of Mn. We require the following conditions.

• Each Dn is a Boolean algebra under the usual set theoretic opera-tions,• Each Dn is invariant under coordinate permutations. That is, if σ :{1, . . . , n} → {1, . . . , n} is a permutation and X ∈ Dn, then σ(X) :={(x1, . . . , xn) ∈ Mn : (xσ−1(1), . . . , xσ−1(n)) ∈ X} ∈ Dn

• For each a ∈ M the singleton set {a} is definable.• The diagonal ∆M := {(x, y) ∈ M2 : x = y} is definable.• The class of definable sets is closed under products: if A ∈ Dn and

B ∈ Dm, then A× B is definable.• The class of definable sets is closed under projections: if A ∈ Dn+1

and π : Mn+1 → Mn is given by (x1, . . . , xn+1) 7→ (x1, . . . , xn), thenπ(A) ∈ Dn.

Remark 3.2. One might prefer to present a structure by giving a collectionBn of basic definable sets X ⊆ Mn and then closing the conditions specifiedin Definition 3.1.

Example 3.3. If M is an algebraically closed field and Dn is the collection ofall constructible subsets of Mn, then M is a structure.


Example 3.4. If G is a group and we ask for the set Γ := {(x, y, z) ∈ G3 : xy =z} to be definable, then many other sets will be definable. For example, thecenter of G, Z(G) := {x ∈ G : (∀y ∈ G)xy = yx} Let see that this is in factthe case. By hypothesis Γ ∈ D3. Thus,

A1 := {(x1, x2, x3, y1, y2, y3) : x1x2 = x3&y1y2 = y3} = Γ× Γ ∈ D6

Applying the condition about closure under coordinate permutations,we see that

A2 := {(x1, x2, y1, y2, x3, y3) : x1x2 = x3&y1y2 = y3} ∈ D6

Since D4 is a Boolean subalgebra of the power set of G4, we see thatG4 ∈ D4. By our condition on the diagonal, ∆ ∈ D2. Hence, since the classof definable sets is closed under products, G4 × ∆ ∈ D6. By the closureunder intersections, we see that

A3 := A2∩ (M4×∆) = {(x1, x2, y1, y2, x3, y3) : x1x2 = x3&y1y2 = y3&x3 = y3} ∈ D6

is definable. Projecting onto the first five coordinates and then onto the firstfour coordinates, we see that the following set is definable.

A4 := {(x1, x2, y1, y2) : x1x2 = y1y2}

Applying coordinate permutation again, we see that the following set isdefinable.

A5 := {(x1, y2, x2, y1) : x1x2 = y1y2}Hence, the following set is definable.

A6 := A5 ∩ (∆× ∆) = {(x1, y2, x2, y1) : x1x2 = y1y2&x1 = y2&x2 = y1}

Applying another permutation, we have that the following set is definable.

A7 := {(x1, x2, y1, y2) : x1x2 = y1y2&x1 = y2&x2 = y1}

Projecting onto the first three coordinates and then onto the first two coor-dinates, we see that the following set is definable.

A8 := {(x1, x2) : x1x2 = x2x1}

Finally, if π : G2 → G is the projection onto the first coordinate, then wesee that Z(G) = G r π(G2 r A8). Thus, Z(G) ∈ D1.

I hope that the calculation in the above example convinces the readerthat the presentation of the class of definable sets of Definition 3.1 is notwell suited to mathematical arguments. Indeed, in checking that a partic-ular set is definable in a given structure, the most effective method is tosimply present the definition. This brings us back to the the definition of astructure as an interpretation of a formal language.

10 THOMAS SCANLON

Definition 3.5. A (first-order) language L is given by the data of a set Cof constant symbols, a sequence of sets 〈Fn〉∞n=1 of function symbols, and asequence of sets 〈Rn〉∞n=1 of relation symbols. An L-structure M is a non-empty set M given together with interpretations of the symbols of L. Thatis, for each c ∈ C are given an element cM ∈ M, for each f ∈ Fn we aregiven a function fM : Mn → M, and for each R ∈ Rn we are given a subsetRM ⊆ Mn.

Remark 3.6. One may recover a structure in the sense of Definition 3.1 froman L-structure in the sense of Definition 3.5 by taking the graph of each fM

and each set RM to be definable and then closing off the class of definablesets under the requirements of Definition 3.1.

Definition 3.7. An o-minimal structure is a structure (R,<, . . .) in the senseof first-order logic for which < is a linear order on R and the ellipses indi-cate that there may be other distinguished functions and relations for whichevery definable (with parameters) subset of R is a finite union of points andintervals.

To understand the conclusion of the Pila-Wilkie counting theorem, it suf-fices to consider only the case where the underlying ordered set is the set ofreal numbers with its usual ordering. However, the proof of that theoremuses more general o-minimal structures in an essential way.

While in Definition 3.5 we allowed for the formal language to have ex-tra distinguished relations and constants, when studying o-minimal struc-tures, the only extra relation required is the ordering. That is, it sufficesto consider only those expansions specified by naming some distinguishedfunctions.

Definition 3.8. Suppose that for each natural number n we are given aset Fn of functions f : Rn → R. By RF we mean the structure whoseunderlying set is R, whose order is the usual order on the real numbers,and which has the distinguished function f : Rn → R for each f ∈ Fn.

Given such a structure RF by a basic or atomic definable set in Rn wemean a set of the form

{(x1, . . . , xn) ∈ Rn : f (x1, . . . , xn) = g(x1, . . . , xn)}or

{(x1, . . . , xn) ∈ Rn : f (x1, . . . , xn) < g(x1, . . . , xn)}where f and g are functions of n variables built from the coordinate func-tions, constant functions and the distinguished functions in F via appro-priate compositions. It follows that the class of definable sets in RF is gen-erated from these basic sets by the rules outlined in Definition 3.1.

Perhaps, some examples are in order.

Example 3.9. Taking F to consist of all polynomials in any number of vari-ables over R, it follows from work of Tarski on the decidability of Eu-clidean geometry [24] that every definable set is semi-algebraic, that is, a


finite Boolean combination of sets defined by by conditions of the formf (x1, . . . , xn) > 0 where f ∈ R[x1, . . . , xn]. Since a polynomial in one vari-able changes sign only finitely many times, it follows that RF is o-minimal.

Example 3.10. Note that {x ∈ R : sin(x) = 0} = Zπ is an infinite, discreteset and as such cannot be expressed as a finite union of points and intervals.Hence, R{sin} is not o-minimal.

Example 3.11. We say that f : Rn → R is a restricted analytic function if thereis a neighborhood U ⊇ [−1, 1]n of the n-cube [−1, 1]n and a real analyticfunction f : U → R for which f (x) = f (x) for x ∈ [−1, 1]n and f (x) = 0for x ∈ Rn r [−1, 1]n. If we let F consist of all polynomials over R andall restricted analytic functions, then van den Dries observed [28] that theo-minimality of RF (usually denoted as Ran) follows as a consequence ofresults of Gabrielov [6] on semi-analytic geometry. Thereafter, Denef andvan den Dries [3] presented a more direct proof of the o-minimality of Ran.The key technical observation required for their proof is that the Weier-strass Preparation and Division Theorems permit one to replace conditionson the sign of an analytic function of a single variable over a closed intervalwith the same conditions on an associated polynomial.

Example 3.12. Extending work of Khovanski on so-called fewnomials [7],Wilkie [31] showed that if F contains all the polynomials over R togetherwith the real exponential function, then Rexp := RF is o-minimal. Wilkieextended this result to obtain the stronger theorem that the expansion of thereal field by all functions which satisfy iterated Pfaffian differential equa-tions is o-minimal [32].

Example 3.13. Amalgamating the last two examples so that F consists ofall restricted analytic functions, all polynomials, and the real exponentialfunction we obtain Ran,exp which van den Dries and Miller proved to be o-minimal [30]. In subsequent work, van den Dries, Macintyre, and Markeranalyzed the definable sets in Ran,exp through the study of generalizedpower series models [29]. Thereafter, Speissegger showed that if RF is ano-minimal structure and f : R → R is a function which satisfies a Pfaf-fian differential equation over RF , that is, there is some G(x, y) ∈ F2 forwhich f satisfies the differential equation Y′ = G(x, Y), then the structureobtained by adjoining f to F1 is still o-minimal [23].

The great virtue of the notion of o-minimality is that from the hypothesisabout the simplicity of the definable subsets of the line one may deducestrong regularity results about the definable sets in higher dimensions. Thefundamental theorem of o-minimality is the cell decomposition theoremwhich was first proven by van den Dries under the hypothesis that theunderlying ordered set is (R,<) [27] and in full generality by Knight, Pillayand Steinhorn [8, 17, 18].

Definition 3.14. Given an o-minimal structure (R,<, . . .) we define theclass of cells in Rn and their dimension by recursion on n. When n = 1,

12 THOMAS SCANLON

singleton sets {a} and intervals (a, b) where we allow the possibility thata = −∞ and that b = ∞ are cells. Their dimensions are 0 and 1, respec-tively. If X ⊆ Rn is a cell and f : X → R is a continuous (with respectto the order topology), definable function (in the sense that its graph is adefinable set), then the graph of f is a cell in Rn+1 with the same dimensionas that of X. If g : X → R is another continuous, definable function on Xfor which f (x) < g(x) for every x ∈ X, then the parametrized interval

( f , g)X := {(x, y) ∈ Rn × R : x ∈ X & f (x) < y < g(x)}is a cell of dimension one more than that of X. Likewise, infinite inter-vals (−∞, f )X and ( f , ∞)X (with the obvious definitions) are cells also ofdimension one more than that of X.

Theorem 3.15. If (R,<, . . .) is an o-minimal structure and X ⊆ Rn is definable,then there is a partition of Rn into finitely many cells so that X may be expressedas a union of some of these cells.

Proof. One argues by induction on n noting that the case of n = 1 is exactlythe definition of o-minimality and proving along the way two auxiliaryresults. First, if f : R → R is any definable function then R may be decom-posed into finitely many points and open intervals so that on each suchopen interval f is strictly monotone or constant. Secondly, every definablefunction f : Rn → R is piecewise continuous in the sense that the domainadmits a decomposition into finitely many cells for which the restriction off is continuous. For the inductive argument one shows first that for everydefinable set X in Rn+1 there is a cell decomposition of Rn+1 compatiblewith X and then establishes the piecewise continuity of definable functionsf : Rn+1 → R.

The key to proving piecewise monotonicity of functions of a single vari-able is the observation that the sets where f is locally increasing (or de-creasing or constant) are definable. Using o-minimality, one shows that ifthe lemma failed, then there would be an open interval I on which f isnever locally constant, locally increasing, or locally decreasing and thenderives a contradiction to o-minimality by considering sets of the form{x ∈ I : f (x) < f (c)} for some fixed c ∈ I.

Piecewise continuity of definable functions f : Rn+1 → R is shown byobserving that the set of points at which f is continuous is definable andthen invoking cell decomposition to conclude that if the result were falsethere would be an open cell on which f is everywhere discontinuous. Onethen reaches a contradiction by considering the family of functions (ga :Rn → R)a∈R given by ga(x1, . . . , xn) := f (x1, . . . , xn, a) which we knowto be piecewise continuous and the functions (hb : R → R)b∈Rn given byh(x) := f (b, x) which we know to be piecewise monotone.

Finally, for X ⊆ Rn+1 a definable set, we define a sequence of (possiblypartial) functions fm : Rn → R by sending a to the mth point in the bound-ary of Xa := {y ∈ R : (a, y) ∈ X}. Via a nontrivial argument one shows


that the cardinality of the boundary of Xa is bounded. By induction, wemay decompose Rn into cells on which all of the functions fi are continu-ous and the truth value of conditions of the form fi(x) ∈ Xx or y ∈ Xx forsome y ∈ ( f j(x), f j+1(x)) is constant. The cell decomposition statement forX follows. �

It it hard to overstate the strength of the geometric consequences of thecell decomposition theorem and its refinements. For example, it implies akind of infinitesimal rigidity on the topology of definable sets living in adefinable family.

It is a fairly easy consequence of the cell decomposition theorem appliedto the total space of a definable family that given a definable family {Xb}b∈Bof definable sets, the cells required for the cell decompositions of the var-ious fibres also vary in definable families. It follows from this uniformitytheorem that at least when the underlying ordered set is the set of real num-bers with its usual ordering that the topology of the sets in a definable fam-ily is rigid.

Proposition 3.16. If {Xb}b∈B is a definable family of definable sets in some o-minimal structure on the real numbers (with the usual ordering) then there areonly finitely many homeomorphism types represented in the family.

As a corollary of Proposition 3.16 we obtain a theorem of Khovanski [7]on fewnomials. To be fair, while the theorem on fewnomials which we shalldiscuss is logically a consequence of Proposition 3.16 both temporally andintellectually it is prior. Khovanski’s work on fewnomials inspired much ofthe development of theory of o-minimality and many of his specific resultsunderly Wilkie’s proof of the o-minimality of Rexp. Moreover, the argu-ment we outline below is patterned on Khovanski’s own proof through thepassage from polynomials of indeterminate degree to exponential polyno-mials.

Theorem 3.17. For fixed integers k and n there are only finitely many homeomor-phism types amongst the following sets

{(a1, . . . , an) ∈ (R+)n :

k

∑i=1

fiami,11 · · · ami,n

n = 0}

as ( f1, . . . , fk) ranges through Rk and m ranges through the k by n matrices withnatural number coordinates.

To prove Theorem 3.17 we observe that it suffices show that there areonly finitely many homeomorphism types even if we allow m to rangethrough Mk×n(R) rather than merely Mk×n(N). The above family of semi-algebraic sets may be embedded into the following Rexp-definable family.

{(a, f , m) ∈ (R+)n × (Rk × (Rn)k) :

k

∑i=1

fi

n

∏j=1

exp(mi,j ln(ai)) = 0}

14 THOMAS SCANLON

The finiteness of the number of homeomorphism types is now a specialcase of Proposition 3.16.

Another crucial property of o-minimal structures is the existence of de-finable choice functions.

Theorem 3.18. Let (R,<,+, 0, 1, · · · ) be an o-minimal expansion of an orderedgroup (where 1 > 0) and {Xb}b∈B a definable family of nonempty definable subsetsof Rm for some m. Then there is a definable function f : B → Rm so that f (b) ∈Xb for each b ∈ B.

Proof. We work by induction on m. If m = 1, then define

f (b) :=

0 if Xb = Ra if a = min Xb

a− 1 if a is the least boundary point and (−∞, a) ⊆ Xb

a + 1 if a = inf Xb and (a, ∞) = Xba+c

2 if a /∈ Xb, a = inf Xb,and c is the least boundary point of Xb greater than a

Let π : Rm+1 → Rm be the projection to the first m coordinates. Thatby our inductive hypothesis, there is a choice function g : B → Rm forthe family of definable sets {π(Xb)}b∈B. For b ∈ B define Yb := {x ∈R : (g(b), x) ∈ Xb}. Then {Yb}b∈B is a definable family of nonemptysubsets of R. Hence there is a choice function h : B → R for this family.Our desired f is given by b 7→ (g(b), h(b). �

4. PILA-WILKIE COUNTING THEOREM

The proof of the Pila-Wilkie counting theorem consists of two majorparts. First, a general theorem about parametrization of definable sets ino-minimal structures is proven. With this result, it is shown that everybounded set may be expressed as the image of a finite number of smoothfunctions on unit boxes having small derivatives. The crucial part of thistheorem is that the finite number and the bounds on the derivatives maybe obtained uniformly in families. Once this result is established, standardmethods from the theory of diophantine approximation may be used tobound the number of rational points in definable sets.

The Pila-Wilkie counting theorem only really makes sense for o-minimalstructures on the real numbers. As such, it may seem that we should (or, atleast, could) restrict attention to such structures in its proof. However, forour applications we require a uniform version of the parametrization the-orem which will follow from a case by case version proven in an arbitraryo-minimal structure. This alone might not be enough to justify the moveto general o-minimal structures as one might imagine that proving the uni-form version over the real numbers might be simply a matter of keepingtrack of variation in families. However, as the reader will see soon enough,our inductive argument will be complicated enough, but if we were to try


to implement it uniformly, it would degenerate into an incomprehensiblemess. Of course, there may be a natural way to retool the argument so asto avoid nonstandard models, but I believe that the conceptual simplifi-cation afforded by this move amply rewards the decision to work modeltheoretically.

Notation 4.1. Throughout this section, M is an o-minimal structure ex-panding an ordered field.

Definition 4.2. Let n ∈ Z+ be a positive integer. We say that a set X ⊆Mn is strongly bounded if there is a positive integer N ∈ Z+ so that X ⊆[−N, N]n. We say that a function is strongly bounded if its graph is stronglybounded.

Remark 4.3. If the underlying ordered field of M is simply the field of realnumbers, then there is no distinction between bounded and strongly bounded.The difference becomes apparent only for nonstandard models.

Notation 4.4. Let ` ∈ Z+ be a positive integer and φ : (0, 1)` → M adefinable function. For α ∈ N` an multi-index, we denote by φ(α) thefunction ∂|α|

∂xα11 ···∂x

α``

φ. Note that while φ(α) might not be defined at every

point in (0, 1)`, by the cell decomposition theorem it is defined on a denseopen definable subset. More generally, if n ∈ Z+ is any positive integerand ψ : (0, 1)` → Mn is a definable function expressed as ψ(x1, . . . , x`) =

(φ1(x1, . . . , x`), . . . , φn(x1, . . . , x`)), then we write φ(α) for (φ(α)1 , . . . , φ

(α)n ).

Definition 4.5. Let X ⊆ Mn be a definable set with dim(X) = `. A partialparametrization of X is a definable function φ : (0, 1)` → X. A parametriza-tion of X is a finite set S of partial parameterizations for which X =

⋃φ∈S φ((0, 1)`).

For a given positive integer k ∈ Z+ we say that the partial parametriza-tion φ : (0, 1)` → X is a partial k-parametrization if φ is Ck and for eachmulti-index α ∈ N` with |α| ≤ k the map φ(α) is strongly bounded. A k-parametrization of X is then a parametrization of X by partial k-parameterizations.

Remark 4.6. In the definition of a k-parametrization, at the cost of increasingthe finite number of functions in S we could strengthen the conclusion fromφ(α) being strongly bounded to |φ(α)| ≤ 1. Indeed, suppose that N ∈ Z+

were an upper bound for |φ(α)|. For a = (a1, . . . , a`) ∈ {0, 1, . . . , N − 1}`define φa(x) := φ( x1+a1

N , . . . , x`+a`N ). A simple calculation using the chain

rule shows that |φ(α)a | ≤ 1 and clearly the image of φ is the union over the

N` such choices of a of the the images of φa.

Definition 4.7. We call a k-parametrization S of X a strong k-parametrizationif for each φ ∈ S and each multi-index α with |α| ≤ k we have |φ(α)(x)| ≤ 1.

The main structural theorem of this section is the following parametriza-tion theorem.

16 THOMAS SCANLON

Theorem 4.8. For any strongly bounded definable set X ⊆ Mn and any positiveinteger k ∈ Z+ there is a k-parametrization of X.

Remark 4.9. By Remark 4.6 we may strengthen Theorem 4.8 to the con-clusion that X admits a strong k-parametrization. It will be convenientthroughout the course of our inductive argument to allow for the relaxednotion of a mere k-parametrization.

I regard Theorem 4.8 as a dual version of the cell decomposition theorem.With the cell decomposition theorem we take an arbitrary definable set Xin Mn and cut up the ambient space into finitely many simple definablepieces, namely the cells, so that X and its complement may be decomposedinto these parts. By bootstrapping the cell decomposition procedure wemay ask that the boundaries of the cells be given by the graphs of arbitrar-ily smooth functions. With Theorem 4.8 instead of breaking X into simplepieces, each of which is definably homeomorphic to some open box, wecover X by the images of such boxes under functions with small deriva-tives. As we shall see, not only are the statements of these theorems analo-gous, but their proofs share a common structure (though to be honest thereis a substantial difference in the organization of the induction to improvethe order of differentiability which we shall address in due course). In or-der to carry out an inductive argument, we simultaneously prove a theo-rem about the regularity of definable functions with an especially strongform of the regularity theorem for functions of a single variable. In the caseof the cell decomposition theorem, this regularity theorem takes the formof the theorems that in dimension one every unary definable function ispiecewise monotone and in general that every definable function is piece-wise continuous where the pieces in question are cells. For Theorem 4.8,the regularity theorem concerns reparameterizations of definable functions.

Definition 4.10. Let X ⊆ Mm be a definable set, Φ : X → Mn a de-finable function and S a k-parametrization of X. We say that S is a k-reparametrization of Φ if for each φ ∈ S the function Φ ◦ φ is Ck and for eachmulti-index α with |α| ≤ k the function (Φ ◦ φ)(α) is strongly bounded.

Remark 4.11. The definition of a k-reparametrization of Φ is almost equiva-lent to asking that {Φ ◦ φ : φ ∈ S} be a k-parametrization of the image of Xunder Φ. The difference appears when Φ collapses dimensions in the sensethat the image of X under Φ has dimension strictly less than that of X.

The reparametrization theorem takes the following form.

Theorem 4.12. For every positive integer k every strongly bounded definablefunction admits a k-reparametrization.

Using the compactness theorem one shows that a uniform version ofTheorem 4.8 follows as a corollary. Let us recall the notions of definablefamilies of sets and of functions.


Definition 4.13. By a definable family {Xb}b∈B of definable subsets of Mn

we mean a definable set X ⊆ Mn × B ⊆ Mn × Mm (where B ⊆ Mm isitself a definable set) where for b ∈ B the set Xb is the fibre Xb := {x ∈Mn : 〈x, b〉 ∈ X}. By a definable family of definable functions we meansimply a definable family of definable sets each of which is a definable func-tion, or if your prefer, the graph of a definable function.

Theorem 4.14. For any triple of positive integers m, n, k ∈ Z+ and definablefamily {Xb}b∈B of definable subsets of (0, 1)m there exists a number N ∈ Z+ andN families of definable functions {φi,b}b∈B for i ≤ N so that for each b ∈ B the set{φi,b : i ≤ m} is a strong k-parametrization of Xb.

Proof. To deduce Theorem 4.14 from Theorem 4.8 we use a standard appli-cation of the compactness theorem of first-order logic.

We first prove an apparently weaker result. Given a definable family{Xb}b∈B of definable subsets of (0, 1)m there is some N ∈ Z+ and N fami-lies {φi,c}c∈C of definable functions so that for any b ∈ B there is some I ⊆{1, . . . , N} and some c ∈ C so that {φi,c}i∈I is a strong k-parametrization ofXb.

The full uniform theorem follows from this ostensibly weaker result bythe existence of definable choice functions, Theorem 3.18.

If the uniform theorem were false, there would be some definable fam-ily of definable subsets {Xb}b∈B of (0, 1)m so that for any finite sequenceof definable families of definable functions {φi,c}c∈C for 1 ≤ i ≤ N, therewould be some b ∈ B for which no collection {φi,c : i ∈ I} is a strongk-parametrization of Xb. Applying the compactness theorem to the theoryof R together with the set of sentences (using the new constant symbol b)asserting that for each such family of definable functions there is no pa-rameter c for which {φi,c : 1 ≤ i ≤ n} is a strong k-parametrizationof Xb, we find an o-minimal structure M∗ and a point b ∈ B(M∗) forwhich the strongly bounded definable set Xb does not admit a strong k-parametrization, but this contradicts Theorem 4.8. �

As we indicated above, the proof of Theorem 4.8 is organized as an in-duction with which we interweave a proof of Theorem 4.12 starting with aparticularly strong form of Theorem 4.12 in dimension one. Let us outlinethe structure of the proof. For n, m, k ≥ 1 consider the following assertions.

B(k) If F : (0, 1)→ M is a definable function, then there is a k-reparametrizationof F so that for each φ in the reparametrization either φ or F ◦ φ is apolynomial.

R(m,n,k) Every strongly bounded function F : (0, 1)n → Mm admits a k-reparametrization.

P(n,k) Every strongly bounded definable set X ⊆ Mn admits a k-parametrization.Note that B(k) is a refined form of R(1,1,k) and that Theorem 4.8 is the as-

sertion that for every n and k the condition P(n,k) holds while Theorem 4.12is the assertion that for every n, m and k the condition R(n,m,k) holds.

18 THOMAS SCANLON

One begins with the observation that P(1,k) is a trivial consequence ofthe definitions. One then proves B(k) by induction on k (thereby estab-lishing R(1,1,k) for all k). For the base case of B(1) one considers a simplelinear change of variables. The inductive case uses a trick thereby improv-ing the number of derivatives which are strongly bounded through a qua-dratic change of variables. Through an induction on n one shows that toprove R(m,n,k) it suffices to prove R(1,n,k). One then completes core ofthe inductive argument by showing (∀k)P(n, k) → (∀k)R(n + 1, 1, k) and(∀k, n)R(m, n, k)→ (∀k)P(n + 1, k).

With Theorem 4.14 established, we deduce the counting theorem. Webegin with a proposition about algebraic relations on the rational points inthe image of a function with small derivatives.

Proposition 4.15. Given m < n, and d ∈ Z+ there is a positive integer k andpositive constants ε = ε(m, n, d) and C = C(m, n, d) so that if φ : (0, 1)m →Rn is a Ck function with image X satisfying | ∂|α|

∂xα11 ···∂xαm

mφi(x)| ≤ 1 for all multi-

indices α with |α| ≤ k, then for any t ≥ 1 the set X(t) is contained in the unionof at most ctε (not necessarily irreducible) hypersurfaces of degree d. Moreover,ε(m, n, d)→ 0 as d→ ∞.

Proof. We begin by changing the problem slightly observing that it wouldbe enough to show that the set of rational points with a fixed denominatort lie in O(tε) hypersurfaces of degree d.

Take Q1, . . . , Q` ∈ (0, 1)m so that the points φ(Qi) are the distinct rationalpoints in X with denominator t. Let D := (n+d

d ). Consider the `×D-matrix(φ(Qi)

µ) where µ ranges through the multi-indices in N` with |µ| ≤ d.Using a Taylor series expansion and our presumed bound on the size of thederivatives of φ, we see that there is a constant b = b(m, n, d) ∈ Z+ andC′ > 0 so that for any 0 < r < 1 if z1, . . . , zD are points in (0, 1)m all withina given ball of radius r, then |det(φ(zi))

µ| ≤ C′rb. On the other hand, if thezi’s are chosen from the Qj’s then det(φ(zi)

µ) ∈ 1tD Z. Thus, if they are all

taken from the same small enough ball, this determinant must be zero. Itfollows from the implied linear dependencies on the columns of the matrix(φ(zi)

µ) that when the points Qj are all taken from the same small ball, thentheir images lie on a common hypersurface of degree d. Covering (0, 1)m

with small enough balls, we see that the set of rational points on X withfixed denominator t is contained in (C′)

nb tDnb hypersurfaces of degree d.

To complete the argument one needs to be somewhat more careful with thecalculation of the combinatorial quantities D and b than I have been, butupon doing so one sees that the exponent Dn

b tends to zero as d→ ∞. �

Combining Proposition 4.15 with Theorem 4.14 we deduce a qualitativeversion of the counting theorem.

Proposition 4.16. Let {Xb}b∈B be a definable family of subsets of (0, 1)n andε > 0. Then there is a constant C > 0 and a number d ∈ Z+ depending only on


the family and ε so that for any b ∈ B and t ≥ 1, the set Xb(t) is contained in nomore than Ctε hypersurfaces of degree d.

Proof. Let k and d be large enough so that the quantity ε(m, n, d) of Propo-sition 4.15 is less than our prescribed ε. By Theorem 4.14, there is a num-ber M so that every Xb may be covered by the image of at most M strongk-parametrizations. The constant C of this proposition is M times the con-stant of Proposition 4.15. �

The counting theorem itself now follows from Proposition 4.16 by induc-tion on the fibre dimension of Xb.

5. APPLICATIONS

In this section we sketch some of the more sophisticated diophantinetheorems proved using the Pila-Wilkie counting theorem.

We begin with the first nontrivial instance of the Zilber-Pink conjecturein which one considers intersections with positive dimensional special va-rieties rather than merely with special points: a theorem of Masser andZannier [11] about simultaneous torsion on elliptic curves. They considerthe family of elliptic curves presented in their affine Legendre form whereEλ is defined by the affine planar equation y2 = x(x − 1)(x − λ) for λ ∈C r {0, 1}. From theory of elliptic curves, Eλ considered together with thepoint at infinity has a unique structure of an algebraic group with that pointat infinity as the identity. For a fixed complex number a we might con-sider the set of λ for which the point (a,

√a(a− 1)(a− λ)) is torsion in the

group Eλ(C). It is not hard to see that for a = 0 or a = 1, then these pointsare always torsion. On the other hand, for every other a there are onlycountably many λ for which this point is torsion in Eλ(C). Nevertheless,computing the rational functions which define the multiplication by n mapon Eλ it is fairly easy to show that for any such a there will be infinitelymany λ for which (a,

√a(a− 1)(a− λ)) is torsion. Masser and Zannier ad-

dress the question: if we consider two number a and b, for how many λ are(a,

√a(a− 1)(a− λ)) and (b,

√b(b− 1)(b− λ)) both torsion in Eλ(C)? In

the special case of a = 2 and b = 3 they given an answer.

Theorem 5.1. There are only finitely many complex numbers λ for which

Pλ := (2,√

2(2− λ))

andQλ := (3,

√6(3− λ))

are torsion in Eλ(C).

Remark 5.2. The proof of Theorem 5.1 applies perfectly well to any twonumbers a and b for which the points

Paλ := (a,

√a(a− 1)(a− λ))

20 THOMAS SCANLON

and

Qbλ := (b,

√b(b− 1)(b− λ))

are linearly independent over Z in the group Eλ(Q(λ)). Indeed, in [10]the same authors prove such a result, even allowing for the sections to benonconstant.

The proof of Theorem 5.1 follows the pattern of the proof of Theorem 2.3we have outlined above. For each elliptic curve Eλ, the the theory of an-alytic uniformizations gives a complex analytic covering map πλ : C →Eλ(C). As with the usual exponential function, this covering is not de-finable in any o-minimal expansion of the real numbers. However, if werestrict πλ to a fundamental domain, it is. Moreover, at the cost of treatingπλ as simply a real analytic function, we may normalize the fundamentaldomain so that the domain of πλ is the square [0, 1) × [0, 1) and the mapπλ is a group homomorphism when [0, 1) is given the usual wrap aroundadditive structure. With some work, one can show that the two variable(or, really, four real variable) function (λ, z) 7→ πλ(z) is definable in Ran,exprelative to the usual interpretation of C in R and when z is restricted to[0, 1)× [0, 1). Masser and Zannier then study the set

X := {(x1, y1, x2, y2) ∈ [0, 1)4 : (∃λ)πλ(x1, y1) = (2,√

2(2− λ))

& πλ(x2, y2) = (3,√

6(3− λ))}

Visibly, X is definable in Ran,exp and it is not hard to see that the ratio-nal points on X all come from λ for which Pλ and Qλ are simultaneouslytorsion. Transcendence results about the Weierstraß ℘-function are used inplace of Ax’s theorem to show that Xalg is empty and a theorem of David [2]about the degree of the field extension required to define elliptic curveswith elements of specified order plays the role of the calculation of the de-gree of a cylcotomic extensions.

It bears noting that the published sketch of Theorem 5.1 avoids an ex-plicit reference to definability in o-minimal structures as Pila had proved aprovisional version of Theorem 2.9 for subanalytic surfaces without invok-ing the theory of o-minimality [13]. On the other hand, due to the work ofPeterzil and Starchenko [12] on the uniform definability of theta functionsin Ran,exp, it follows that the question of simultaneous torsion in families ofhigher dimensional abelian varieties may be analyzed via these methods.

Finally, let us close with Pila’s proof of the Andre-Oort conjecture formodular curves. We shall introduce the Andre-Oort conjecture via the clas-sical theory of complex elliptic curves. Unlike most other approaches tothis problem where one might (or might not) define the terms using com-plex analysis but then address the questions with a more number theoretictheory, Pila’s proof appeals directly to the complex analytic presentation ofthe problem.


As we observed above, for every elliptic curve E over the complex num-bers, one can find a complex analytic surjective group homomorphism π :C → E(C). The kernel of C is a lattice which after making a linear changeof variables we may express as ker π = Z⊕Zτ for some complex numberτ ∈ h := {z ∈ C : Im(z) > 0}. Conversely, for any τ ∈ h, the complexanalytic group Eτ(C) := C/(Z + Zτ) is complex analytically isomorphicto a complex algebraic curve with an algebraic group structure, which weshall continue to denote by Eτ. From the general theory of covering spaces,it is not hard to see that the endomorphisms of the elliptic curve Eτ cor-respond to complex numbers µ for which µ(Z + Zτ) ≤ Z + Zτ. A shortcomputation shows that for most choices of τ, the number µ gives an en-domorphism only when µ is an integer. On the other hand, if τ satisfiesa quadratic equation over Q, then there will be some endomorphisms notcoming from Z. This is the reason why elliptic curves whose endomor-phism rings are strictly larger than Z are said to have complex multiplicationor to be CM.

There is an analytic function j : h → C having the property that Eτ(C)and Eσ(C) are isomorphic as elliptic curves if and only if j(τ) = j(σ). Werefer to the value j(τ) as the j-invariant of the elliptic curve Eτ. Let us saythat a complex number ζ is a special point if it is the j-invariant of an ellipticcurve with complex multiplication. By the above discussion, we see that anumber is special if and only if it is the value of the analytic j-function on aquadratic imaginary number. The Andre-Oort conjecture in this case pre-dicts the form of the algebraic subvarieties X ⊆ An

C of affine n-space whichcontain a Zariski dense set of n-tuples of special points. Specializing to thecase of n = 2, it proposes a solution to the question of for which polynomi-als g(x, y) ∈ C[x, y] are there infinitely many pairs (ξ, ζ) of special pointsfor which g(ξ, ζ) = 0? This case was solved early in the investigationsaround the Andre-Oort conjecture, first assuming the Riemann Hypothesisby Edixhoven [4] and then unconditionally by Andre [1].

Clearly, if ξ is a special point, then the set algebraic varieties {ξ} ×A1C

and A1×{ξ} contain Zariski dense sets of special points as does the wholeplane A2

C. It follows from the general theory of coverings, that for eachn ∈ Z+ there is a polynomial Pn(x, y) ∈ C[x, y] for which the function τ 7→Pn(j(nτ), j(τ)) is identically zero. From this presentation, it is clear that thecurve defined by the vanishing of Pn contains a Zariski dense set of specialpoints for if τ is quadratic imaginary, then so is nτ and vice versa. TheAndre-Oort conjecture (for the j-line) says that these are the only algebraicvarieties other than points which can contain a Zariski dense set of specialpoints.

Theorem 5.3 (Pila). Let X ⊆ AnC be an irreducible algebraic subvariety of affine

n-space over the complex numbers. Suppose that the set

{(ξ1, . . . , ξn) ∈ X(C) : each ξi is the j-invariant of a CM-elliptic curve}

22 THOMAS SCANLON

is Zariski dense in X, then X is defined by equations of the form Pm(xi, xj) = 0and xk = ξ for ξ a special point.

The proof of Theorem 5.3 follows our by now familiar pattern. First, Pilaobserves that j restricted to a fundamental domain is definable in Ran,exp.Indeed, again relative to the usual interpretation of C as R2, the complexexponential function is definable in Rexp when restricted to a fundamen-tal domain for j and maps the fundamental domain into a disk properlycontained in the unit disk. It is well known that the q expansion of thej-function is meromorphic on the unit disk with a simple pole at the ori-gin. Hence, the restriction of this function to any proper subdisk is defin-able in Ran. Thus, the restriction of the j-function to a fundamental do-main is definable in Ran,exp. One then moves from a study of X to thatof X, the inverse image of X(C) via j (or really, the map (z1, . . . , zn) 7→(j(z1), . . . , j(zn))) restricted to its fundamental domain, which is a defin-able set in Ran,exp. One must determine Xalg. In this case, one considersof the action of the modular group showing that the algebraic part comesfrom the pre-images of finitely many varieties of the desired form. At thispoint, the goal is to show that if X does not already have the desired form,then there are only finitely many quadratic imaginary points in Xtr. Thecounting theorem, Theorem 2.9, applied to rational points, but Pila provessimilar bounds for algebraic points of bounded degree [14]. Thus, for anyε > 0 there is some constant C for which the number of quadratic imag-inary points of height at most t in Xtr is at most Ctε. As in the proof ofTheorem 2.3, he reduces to the case that X is defined over a number fieldand observes that if there are special points coming from Xtr, then all oftheir Galois conjugates are also in this set. At this point, he estimates thesize of these orbits from below using Siegel’s theorem on the growth of theclass number [22] to find that for ε < 2 one has a lower bound of Ctε thuscontradicting the upper bound from the counting theorem.

Remark 5.4. Theorem 5.3 had been proven previously by Edixhoven andYafaev [5] under the assumption of the Generalized Riemann Hypothesisfor quadratic imaginary fields. Their proof shares the same kind strategyat the end: find upper bounds geometrically and lower bounds via Galoistheory and analytic number theory.

Remark 5.5. The paper in which the proof of Theorem 5.3 appears [15] in-cludes proofs of theorems in the direction of the Pink-Zilber conjectures.On the other hand, while many parts of this argument succeed when ap-plied to other Shimura varieties, some steps are incomplete. For example,it is known that the analytic covering maps for the moduli spaces of prin-cipally polarized abelian varieties are definable in Ran,exp (again, after suit-able restriction) [12] and it seems plausible that the arguments employedto determine the algebraic parts of inverse images of algebraic varieties byCartesian powers of j should work for these maps, too, but to date no one


has carried out the details. Recent work of Tsimerman [25] and Ullmo andYafaev [26] has established exponential lower bounds for the size of Galoisorbits of special points on some moduli spaces of abelian varieties. It isnot unreasonable to hope that these methods might lead an unconditionalproof of the Andre-Oort conjecture.

REFERENCES

[1] Yves Andre. Finitude des couples d’invariants modulaires singuliers sur une courbealgebrique plane non modulaire. J. Reine Angew. Math., 505:203–208, 1998.

[2] Sinnou David. Points de petite hauteur sur les courbes elliptiques. J. Number Theory,64(1):104–129, 1997.

[3] J. Denef and L. van den Dries. p-adic and real subanalytic sets. Ann. of Math. (2),128(1):79–138, 1988.

[4] Bas Edixhoven. Special points on the product of two modular curves. Compositio Math.,114(3):315–328, 1998.

[5] Bas Edixhoven and Andrei Yafaev. Subvarieties of Shimura varieties. Ann. of Math. (2),157(2):621–645, 2003.

[6] A. M. Gabrielov. Projections of semianalytic sets. Funkcional. Anal. i Prilozen., 2(4):18–30, 1968.

[7] A. G. Khovanskiı. Fewnomials, volume 88 of Translations of Mathematical Monographs.American Mathematical Society, Providence, RI, 1991. Translated from the Russian bySmilka Zdravkovska.

[8] Julia F. Knight, Anand Pillay, and Charles Steinhorn. Definable sets in ordered struc-tures. II. Trans. Amer. Math. Soc., 295(2):593–605, 1986.

[9] Henry B. Mann. On linear relations between roots of unity. Mathematika, 12:107–117,1965.

[10] D. Masser and U. Zannier. Torsion points on families of squares of elliptic curves. Math-ematische Annalen, 16 February 2011. Online First.

[11] D. Masser and U. Zannier. Torsion anomalous points and families of elliptic curves.Amer. J. Math., 132(6):1677–1691, 2010.

[12] Ya’acov Peterzil and Sergei Starchenko. Definability of restricted theta functions andfamilies of abelian varieties. preprint, 2010.

[13] Jonathan Pila. Rational points on a subanalytic surface. Ann. Inst. Fourier (Grenoble),55(5):1501–1516, 2005.

[14] Jonathan Pila. On the algebraic points of a definable set. Selecta Math. (N.S.), 15(1):151–170, 2009.

[15] Jonathan Pila. O-minimality and the Andre-Oort conjecture for Cn. Ann. of Math. (2),(to appear).

[16] Jonathan Pila and Umberto Zannier. Rational points in periodic analytic sets and theManin-Mumford conjecture. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei(9) Mat. Appl., 19(2):149–162, 2008.

[17] Anand Pillay and Charles Steinhorn. Definable sets in ordered structures. I. Trans.Amer. Math. Soc., 295(2):565–592, 1986.

[18] Anand Pillay and Charles Steinhorn. Definable sets in ordered structures. III. Trans.Amer. Math. Soc., 309(2):469–476, 1988.

[19] M. Raynaud. Sous-varietes d’une variete abelienne et points de torsion. In Arithmeticand geometry, Vol. I, volume 35 of Progr. Math., pages 327–352. Birkhauser Boston,Boston, MA, 1983.

[20] Thomas Scanlon. The Andre-Oort conjecture via o-minimality (after J. Pila). 2011.Seminaire Bourbaki. Vol. 2010/2011.

24 THOMAS SCANLON

[21] Thomas Scanlon. Counting special points: logic, diophantine geometry and transcen-dence theory. 8 January 2011. Current Events Bulletin.

[22] C. L. Siegel. Uber die Classenzahl quadratischer Zahlkorper. Acta Arith., (1):83–86,1935.

[23] Patrick Speissegger. The Pfaffian closure of an o-minimal structure. J. Reine Angew.Math., 508:189–211, 1999.

[24] Alfred Tarski. A Decision Method for Elementary Algebra and Geometry. RAND Corpora-tion, Santa Monica, Calif., 1948.

[25] J. Tsimermann. Brauer-siegel for arithmetic tori and lower bounds for galois orbits ofspecial points. preprint, 2011.

[26] E. Ullmo and A. Yafev. Nombre de classes des tores de multiplication complexe etbornes inferieures pour orbites Galoisiennes de points speciaux. preprint, 2011.

[27] Lou van den Dries. Remarks on Tarski’s problem concerning (R, +, ·, exp). In Logiccolloquium ’82 (Florence, 1982), volume 112 of Stud. Logic Found. Math., pages 97–121.North-Holland, Amsterdam, 1984.

[28] Lou van den Dries. A generalization of the Tarski-Seidenberg theorem, and some non-definability results. Bull. Amer. Math. Soc. (N.S.), 15(2):189–193, 1986.

[29] Lou van den Dries, Angus Macintyre, and David Marker. The elementary theory ofrestricted analytic fields with exponentiation. Ann. of Math. (2), 140(1):183–205, 1994.

[30] Lou van den Dries and Chris Miller. On the real exponential field with restricted ana-lytic functions. Israel J. Math., 85(1-3):19–56, 1994.

[31] A. J. Wilkie. Model completeness results for expansions of the ordered field of realnumbers by restricted Pfaffian functions and the exponential function. J. Amer. Math.Soc., 9(4):1051–1094, 1996.

[32] A. J. Wilkie. A theorem of the complement and some new o-minimal structures. SelectaMath. (N.S.), 5(4):397–421, 1999.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY, EVANS HALL,BERKELEY, CA 94720-3840, USA

E-mail address: [email protected]

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