o-minimality and certain atypical intersections

8/11/2019 O-minimality and Certain Atypical Intersections

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arXiv:1409

.0771v1

[math.NT

]2Sep2014

O-MINIMALITY AND CERTAIN ATYPICAL INTERSECTIONS

P. HABEGGER AND J. PILA

Abstract. We show that the strategy of point counting in o-minimal struc-tures can be applied to various problems on unlikely intersections that go

beyond the conjectures of Manin-Mumford and Andre-Oort. We verify theso-called Zilber-Pink Conjecture in a product of modular curves on assuminga lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel

Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Con-jecture for curves unconditionally when everything is defined over a number

field. For higher dimensional subvarieties of abelian varieties we obtain someweaker results and some conditional results.

On demontre que la strategie de comptage dans des structures o-minimalesest suffisante pour traiter plusieurs problemes qui vont au-dela des conjectures

de Manin-Mumford et Andre-Oort. On verifie la conjecture de Zilber-Pinkpour un produit de courbes modulaires en supposant une minoration assez

forte pour la taille de lorbite de Galois et en supposant une version modulairedu theoreme de Ax-Schanuel. Dans le cas des varietes abeliennes on demontrela conjecture de Zilber-Pink pour les courbes si tous les objets sont definis

sur un corps de nombres. Pour les sous-varietes de dimension superieure onobtient quelques resultats plus faibles et quelques resultats conditionnels.

Contents

1. Introduction 22. The Zilber-Pink Conjecture 43. Special subvarieties 73.1. The abelian setting 73.2. The modular setting 94. Geodesic-optimal subvarieties 115. Weak complex Ax 135.1. The abelian setting 135.2. A product of modular curves 146. A finiteness result for geodesic-optimal subvarieties 166.1. The abelian case 166.2. Mobius varieties 207. Counting semi-rational points 22

8. Large Galois orbits 269. Unlikely intersections in abelian varieties 279.1. The arithmetic complexity of a torsion coset 279.2. LGO and the Neron-Tate height 319.3. Intersecting with algebraic subgroups 3510. Unlikely intersections inY(1)n 38Acknowledgements 39References 40

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2 P. HABEGGER AND J. PILA

1. Introduction

The object of this paper is to show that the o-minimality and point-counting

strategy can be applied to quite general problems of unlikely intersection typeas formulated in the Zilber-Pink Conjecture (ZP; see Section 2 for various for-mulations), provided one assumes certain arithmetic and functional transcendencehypotheses. In these problems there is an ambient variety X of a certain typeequipped with a distinguished collection S of special subvarieties. The conjec-ture governs the intersections of a subvariety V Xwith the members ofS. In theproblems we consider,Xwill be either a product of non-compact modular curves(for which it is sufficient to consider the case X= Y(1)n) or an abelian variety, butthe same formulations should be applicable more generally. In this paper, a subva-riety is always geometrically irreducible and therefore in particular non-empty. Acurve is a subvariety of dimension 1.

Our most general results are conditional, but let us state first an unconditionalresult in the abelian setting.

Say X is an abelian variety defined over a field K and K is a fixed algebraicclosure ofK. For any r R we write

X[r] =

codimXHr

H(K)

whereH runs over algebraic subgroups ofX satisfying the dimension condition.

Theorem 1.1. Let X be an abelian variety defined over a number field K andsupposeV X is a curve, also defined overK. IfV is not contained in a properalgebraic subgroup ofX, thenV(K) X[2] is finite.

This theorem is the abelian version of Maurins Theorem [29]. We will see amore precise version in Theorem9.14.

We briefly describe previously known cases of Theorem 1.1 under additionalhypotheses on V or X. Viada [48] proved finiteness for V not contained in thetranslate of a proper abelian subvariety and ifXis the power of an elliptic curve withcomplex multiplication. Remond and Viada [44] then removed the hypothesis onV. This was later generalised by Ratazzi [39]to when the ambient group variety isisogenous to a power of an abelian variety with complex multiplication. Carrizosasheight lower bound [12,13] in combination with Remonds height upper bound[42] led to a proof for all abelian varieties with complex multiplication. Work ofGalateau [18]and Viada[49] cover the case of an arbitrary product of elliptic curves.

More generally we show, in the abelian and modular settings, that the Zilber-Pink conjecture may be reduced to two statements, one of an arithmetic nature,the other a functional transcendence statement. In general, the former statementremains conjectural in both settings. In the abelian setting, the functional tran-

scendence statement follows from a theorem of Ax[3], while in the modular settinga proof of it has been announced recently by Pila-Tsimerman. Both statementsare generalisations of statements which have been used to establish cases of theAndre-Oort conjecture, and this aspect of our work is in the spirit of Ullmo [46].

The arithmetic hypothesis, which we formulate here, is the Large Galois Orbithypothesis (LGO) and asserts that, for fixed V X, certain (optimal) isolatedintersection points of V with a special subvariety T S have a large Galoisorbit over a fixed finitely generated field of definition for V, expressed in terms of a


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O-MINIMALITY AND INTERSECTIONS 3

suitable complexity measure ofT. Special subvarieties in our settings are describedin Section2 for abelian varieties and Section3.2 for Y(1)n and LGO is formulatedin Section8.

In the context of the Andre-Oort Conjecture, the Generalised Riemann Hypoth-esis (GRH) suffices to guarantee LGO (see [45, 47]). However, it is not clear tothe authors if a variant of the Riemann Hypothesis leads to large Galois orbits forisolated points in V T if dim T 1. Indeed, in the Shimura setting, there seemsto be no galois-theoretic description of isolated points in V T which is rootedin class field theory. On the other hand, suitable bounds have been establishedunconditionally for Andre-Oort in several cases, and perhaps LGO will be foundaccessible without assuming GRH.

Associated withX is a certain transcendental uniformisation : U X. Thefunctional transcendence hypothesis is the Weak Complex Ax hypothesis (WCA)and is a weak form of an analogue for of Ax-Schanuel for cartesian powers of theexponential function. The latter result, due to Ax [2], affirms Schanuels Conjecture

(see[24,p. 30]) for differential fields. WCA is formulated in Section 5.In the modular case X=Y(1)n our result is the following. A very special caseof it was established unconditionally by us in [21].

Theorem 1.2. If LGO and WCA hold forY(1)n then the Zilber-Pink Conjectureholds for subvarieties ofY(1)n defined overC. Moreover, if WCA holds forY(1)n

and LGO holds withK= Q, then the Zilber-Pink Conjecture holds for subvarietiesofY(1)n defined overQ.

In the case that Xis an abelian variety, we establish the same result in Section9. However, as mentioned above, in this case WCA is known, and LGO can beestablished in the case that V is one-dimensional when X and V are defined overQ. This allows us to prove the above unconditional result for curves.

All current approaches towards Theorem 1.1 require a height upper bound on

the set of points in question. Like many of the papers cited above we use Remondsheight bound [42]which relies on his generalisation of Vojtas height inequality.

In contrast to previous approaches we do not rely on delicate Dobrowolski-type[12, 13, 39]or Bogomolov-type [18]height lower bounds to pass from boundedheight to finiteness. These height lower bounds are expected (but not known) togeneralise to arbitrary abelian varieties. Instead we will use a variation of the strat-egy originally devised by Zannier to reprove the Manin-Mumford Conjecture [35]for abelian varieties. This approach relied on the Pila-Wilkie point counting resultin o-minimal structures. We will still require a height lower bound. However, therobust nature of the method allows us to use Massers general bound [27] whichpredates the sophisticated and essentially best-possible results of Ratazzi and Car-rizosa that require the ambient abelian variety to have complex multiplication.

In her recent Ph.D. thesis, Capuano [11]counted rational points on suitable de-finable subsets of a Grassmanian to obtain finiteness results on unlikely intersectionswith curves in the algebraic torus.

In the next theorem we collect several partial results in the abelian setting forsubvarieties of arbitrary dimension.

Theorem 1.3. LetV Xbe a subvariety of an abelian variety, both defined overa number field K. Let us also fix an ample, symmetric line bundle onX and itsassociated Neron-Tate heighth.


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(i) IfS 0 then

P V(K) X[1+dimV]; h(P) S

is contained in a finite union of proper algebraic subgroups ofX.(ii) Supposedim (V) = min{dim X/Y, dim V}for all abelian subvarietiesY

X where : X X/Y is the canonical morphism. If dim V 1 thenV(K) X[1+dimV] is not Zariski dense inV.

Theorem9.15refines this statement. A particularly simple example of a surfacethat does not satisfy the hypothesis in (ii) is the square V Vof a curve V X.The Zilber-Pink Conjecture remains open for surfaces in abelian varieties definedover a number field. In Corollary9.9 we show that a sufficiently strong heightupper bound leads to a proof of the Zilber-Pink Conjecture for abelian varietiesover number fields.

The burden of the theorems is that the o-minimality/point-counting strategy is

adequate to deal with atypical intersections, given these additional ingredients.To some extent this is already demonstrated for curves by the work of Masser andZannier [28] and the authors earlier work [21]. Here we will handle subvarietiesof arbitrary dimension and confirm that the o-minimal method scales to this gen-erality. Ullmo[46] shows that a large Galois orbit statement, Ax-Lindemann,and a height upper bound for certain preimages of special points, together with asuitable definability result, enable a proof of the Andre-Oort Conjecture by point-counting and o-minimality. In our setting, the special subvarieties generally havepositive dimension and are often unaffected by the Galois action. Rather we mustcount ob jects that arise when intersecting them with the given subvariety. Moregenerally, one can formulate results along the lines of the Counting Theorem of[34] for atypical intersections of a definable set in an o-minimal structure withlinear subvarieties defined over Q (or indeed the members of any definable family

of subvarieties having rational parameters). Here (as in previous results by thesemethods) o-minimality is used for more than point-counting.

For the basics about o-minimality see[15]. The definability properties requiredin this paper are afforded by the result, due to Peterzil-Starchenko [31], that the

j-function restricted to the usual fundamental domain F0 for the SL2(Z) actionon the upper half plane H is definable in the o-minimal structure Ran,exp (see[16]). Accordingly, in this paper definable will mean definable in Ran,exp unlessstated otherwise. However, the exponential function is superfluous when workingwith abelian varieties. Here it is enough to work in Ran, the structure generatedby restricted real analytic functions, which was recognised as being o-minimal byvan den Dries after work of Gabrielov. In Section 7we will work with more generalo-minimal structures.

2. The Zilber-Pink Conjecture

The Zilber-Pink Conjecture is a far-reaching generalisation of the Mordell-Langand Andre-Oort Conjectures. Different formulations in different settings, but basedon the same underlying idea of atypical or unlikely intersections, were madeby Zilber [52], Pink[36, 37], and Bombieri-Masser-Zannier[10].

Bombieri-Masser-Zannier [8] show that all the versions are equivalent for Gnm.Essentially the same argument shows they are equivalent for Y(1)n, but in the


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general case this is unclear. The version we give is in any case the strongest:essentially it is the Zilber or Bombieri-Masser-Zannier statement in Pinks generalsetting, where the ambient variety is a mixed Shimura variety (see [ 37]). For anintroduction to the conjecture and the state-of-the-the-art see[51].

The general setting involves and ambient variety which is a mixed Shimura va-riety (see[36]). A mixed Shimura variety X is endowed with a (countable) collec-tion S = SX ofspecial subvarietiesand a larger (usually uncountable) collectionW= WX ofweakly special subvarieties. Special subvarieties of dimension zero arecalled special points.

SayXis an algebraic torus Gnm, an abelian variety, or even a semi-abelian varietydefined over C. In some cases Xappears as a special subvariety of a family ofsemi-abelian varieties and may be treated as a mixed Shimura variety. In general,X shares enough formal properties with mixed Shimura varieties to formulate aZilber-Pink Conjecture on unlikely intersections. A coset ofXis the translate of aconnected algebraic subgroup ofX. By abuse of notation we also call cosets weakly

special subvarieties ofX. A torsion coset ofXis a coset ofXthat contains a pointof finite order. We sometimes call torsion cosets special subvarieties ofX. We alsowriteSX andWX for the set of all cosets and torsion cosets ofX, respectively. Sospecial points are torsion points. We study at torsion cosets in more detail in 3.1.

Definition 2.1. LetXbe a mixed Shimura variety or a semi-abelian variety definedover C. Let S be the collection of special subvarieties of X. Let V X be asubvariety. A subvarietyA V is called atypical (forV inX) if there is a specialsubvarietyT ofX such thatA is a component ofV T and

dim A > dim V+ dim T dim X

i.e. A is an atypical component in Zilbers terminology[52]. The atypical set ofV (inX) is the union of all atypical subvarieties, and is denotedAtyp(V, X).

A special subvariety of a mixed Shimura variety is itself a mixed Shimura variety,withST ={A; A is a component ofT Sfor some S SX},

WT ={A; A is a component ofT Sfor some S WX}.

The following is a CIT (cf. Conjecture 2 [52]) version of Pinks Conjecture, andis on its face stronger than the statement conjectured by Pink. It is convenientto frame it as a statement about all special subvarieties of a given mixed ShimuravarietyX.

Conjecture 2.2(Zilber-Pink (ZP) forX). LetXbe a mixed Shimura variety or asemi-abelian variety defined overC. LetT SX andV T. ThenAtyp(V, T)is a

finite union of atypical subvarieties. Equivalently, V contains only a finite numberof maximal atypical (forV inT) subvarieties.

Now some subvarieties are more atypical than others. Since the collection ofspecial subvarieties is closed under taking irreducible components of intersectionsand contains X, given A Xthere is a smallest special subvariety containing Awhich we denote A. We abbreviate P= {P} for a singleton {P}. We define(following Pink [37]) the defect ofA as

(2.1) (A) = dimA dim A.

ThenA Vis atypical for V in X if(A)< dim X dim V.


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The atypical set is simply a union of atypical subvarieties of V, and it mayhappen that an atypical subvariety is contained in some larger but less atypicalsubvariety. A generalisation of the argument showing that ZP implies the Andre-Oort Conjecture (which corresponds to subvarieties of defect 0) shows that ZPimplies a notionally stronger version in which one considers subvarieties of eachdefect separately.

For T SX , V T, and a positive integer denote by

(2.2) Atyp(V, T)

the union of subvarietiesA V with(A) . Note that this does not depend onT since under our assumptionsV T.

Conjecture 2.3(Articulated Zilber-Pink (AZP) forX). LetXbe a mixed Shimuravariety or a semi-abelian variety defined overC. LetT SX ,V T, anda non-negative integer. ThenAtyp(V, T)is a finite union of atypical subvarieties of defect.

Since the components ofVhave defect at most dim X dim V, the conjectureis trivial for dim X dim V. The following implication uses only the formalproperties of special subvarieties; the reverse implication is immediate. A versionof this result appears in[52].

Proposition 2.4. LetXbe as in Conjecture2.3. Then ZP forX implies AZP forX.

Proof. By induction on the dimension of the ambient special subvariety T =V X. AZP is clear if dim T= 0. So we assume now that ZP holds for Twhile AZPholds for all proper special subvarieties ofT. Let V T. By ZP, there are finitelymany atypical subvarieties Ai, with associated special subvarieties Ti, such thatevery atypical subvariety A is contained in some Ai. The Ti are evidently proper

(V Tis not atypical for V in T). Now fix . Then with i = (Ai),Atyp(V, T) =

i:i

Ai

i:i>

Atyp(Ai, Ti)

is a finite union by the induction hypothesis, which gives AZP for the Ai.

Let us formulate one last conjecture in the same spirit as those above.

Definition 2.5. LetXbe a mixed Shimura variety or a semi-abelian variety definedoverC. LetV X be a subvariety. A subvariety A V is said to be optimal ifthere is no subvarietyB withA B V such that

(B) (A).

The specification ofV andX will generally be suppressed, as no confusion shouldarise. We writeOpt(V) for the set of all optimal subvarieties ofV.

It is illustrative to consider some formal properties of optimal subvarieties A Opt(V). Clearly, A is an irreducible component ofV A. The whole subvarietyis always optimal, i.e. V Opt(V). So maximal optimal subvariety is a uselessconcept. In a certain sense maximality is built into the notation of optimality.Indeed, an optimal subvariety of defect 0 is a maximal torsion coset containedcompletely in X. IfA =V, then(A)< (V) or in other words

dim A > dimA + dim V dimV.


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So A is atypical for V in V. We will see that the arithmetically interesting caseis when A = {P} is a singleton. Then P is contained in a special subvariety ofdimension strictly less than dimV dim V.

Conjecture 2.6. Let X be a mixed Shimura variety or a semi-abelian varietydefined overC. ThenXcontains only finitely many optimal subvarieties.

Lemma 2.7. LetX be a mixed Shimura variety or a semi-abelian variety definedoverC. Then the conclusions of Conjectures2.2and2.6are equivalent forX.

Proof. LetVbe a subvariety ofX.First we suppose that Opt(V) is finite. Let T be a special subvariety of X

containingV . We may assume V = T, as Atyp(V, T) = V otherwise. LetA beatypical forV inTand letB A be an optimal subvariety ofV with(B) (A).Then (B)< dim T dim V and so B is also atypical for V in T. Conjecture2.2follows forXas A is contained in a member of the finite set Opt(V){V} of properoptimal subvarieties ofV .

Let us now assume conversely that Conjecture2.2holds forV withT =V. Wemust show that there are only finitely many possibilities for A Opt(V). Clearly,we may assume A V. But then (A) < (V) = dim Tdim V by optimality.So A is also atypical for V in T. It is contained in a subvarietyB that is maximalatypical for V in T. As B comes from a finite set we are done ifA = B . So sayA B . We observe B =V and A lies in Opt(B), which is finite by induction onthe dimension.

A product of modular curves, in particular Y(1)n, is a (pure) Shimura variety.Its special subvarieties are described in detail in Section3.2.

Another example of a (pure) Shimura variety is the moduli space Ag of prin-cipally polarised abelian varieties of dimension g. It is a special subvariety of alarger mixed Shimura varietyXg which consists ofAg fibered at each point by the

corresponding abelian variety.Conjecture2.2 for an abelian varietyXand its special subvarieties is equivalent

to ZP as formulated for subvarieties V X Xg(see[37,5.2] where the equivalenceis proved for Pinks Conjecture; with obvious modifications the argument provesthe equivalence in the version we have given).

Remark 2.8. With the same definitions, the same equivalence presumably holdsfor any weakly special subvariety of a mixed Shimura variety. It seems interestingto ask under which conditions the same holds for a subvariety of a mixed Shimuravariety.

3. Special subvarieties

In the next two sections we discuss in more detail the special subvarieties of anabelian variety and ofY(1)n.

3.1. The abelian setting. In the case of abelian varieties, the special subvarietiesare the torsion cosets, i.e. the irreducible components of algebraic subgroups orin other words, translates of abelian subvarieties by points of finite order. In thissection we recall some basic facts on torsion cosets and we define their complexity.

An inner-product on an R-vector spaceWis a symmetric, positive definite bilin-ear form, : W W R. The volume vol() of a finitely generated subgroup


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ofWwith respect to , is vol() =| det(i, j)|1/2 for any Z-basis (1, . . . , )of . The volume is independent of the choice of basis. The orthogonal complementof a vector subspace UW is U ={w W; w, u= 0 for all u U}.

Let Xbe an abelian variety defined over C with dim X = g 1 and supposethat L is an ample line bundle on X. The degree ofX with respect to L is theintersection number degL X= (L

g[X]) 1.The line bundleL defines a hermitian form

H: T0(X) T0(X) C

on the tangent space T0(X) ofXat the origin. This form is positive definite since

L is ample. It is C-linear in the first argument and satisfies H(v, w) =H(w, v) forv, w T0(X). The real part Re(H) is an inner-product , on T0(X) taken as anR-vector space of dimension 2g. Thus we obtain a norm v= v, v1/2 on T0(X).The imaginary partE= Im(H) is a non-degenerate symplectic form V V R.

Let X T0(X) denote the period lattice ofX. It is a free abelian group ofrank 2g and generates T

0(X) as an R-vector space. Therefore, vol(

X)> 0 where

vol denotes the volume with respect to the inner-product, L. The subgroup Xis discrete in T0(X).

Lemma 3.1. We havedegL X= g!vol(X).

Proof. This is a well-known consequence of the Riemann-Roch Theorem for abelianvarieties, see Chapter 3.6 [6].

Now suppose that Y X is an abelian subvariety of dimension dim Y 1. Thepull-backL|Y ofL by the inclusion mapY Xis an ample line bundle on Y. WetreatT0(Y) as a vector subspace ofT0(X). The hermitian form on T0(Y) inducedbyL|Yis just the restriction ofH. Let Y T0(Y) denote the period lattice ofY.Then degL|YY= (dim Y)!vol(Y) by the previous lemma. The projection formula

implies degL|Y Y = (Ldim Y[Y]). We will abbreviate this degree by degL Y.

The next lemma uses Minkowskis Second Theorem from the Geometry of Num-bers.

Lemma 3.2. There exists a constant c > 0 depending only on (X, L) with thefollowing properties.

(i) There exist linearly independent periods1, . . . , 2dim Y Y withi c degL Y for1 i 2 dim Y and1 2dim Y c degL Y.

(ii) If z X +T0(Y) there exist X with z T0(Y) and z + c degL Y.

Proof. Let 0< 1 . . . 2 dim Ybe the successive minima of Ywith respect tothe closed unit ball {z T0(Y); z 1}. By Minkowskis Second Theorem wehave

(3.1) 1 2 dim Y 22dim Y vol(Y)(2dim Y)

where (n) > 0 denotes the Lebesgue volume of the unit ball in Rn. There existindependent elements 1, . . . , 2dimY Y with i i 2 dim Y. Let =(X, L)> 0 denote the minimal norm of a period in X . Using (3.1) and i we estimate

i 2 dimY 22 dim Y

(2dim Y)2dim Y1vol(Y).


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Two-by-two real matrices with positive determinant act on H by fractional lineartransformations. Ifg GL+2(Q) then the functionsj (z) andj (gz) on H are relatedby a modular polynomial

N(j(z), j(gz)) = 0

for a suitable positive integer N = N(g) (in fact N(g) is the determinant ifg isscaled to have relatively prime integer entries; see[25,Ch. 5, 2]).

Definition 3.5. A strongly special curve inHn is the image of a map of the form

H Hn, z (g1z , . . . , gnz)

whereg1= 1, g2 . . . , gn GL+2(Q).

By astrict partitionwe will mean a partition in which one designated part onlyis permitted to be empty.

Definition 3.6. Let R = (R0, R1, . . . , Rk) be a strict partition of {1, . . . , n} inwhichR0 is permitted to be empty (andk = 0 is permitted). For each index j we

letHRj denote the corresponding cartesian product. A weakly special subvariety (oftypeR) ofHn is a product

Y =

kj=1

Yj

whereY0 HR0 is a point and, for j = 1, . . . , k, Yj is a strongly special curve inHRj . We havedim Y =k.

Definition 3.7. A weakly special subvariety is called a special subvariety if eachcoordinate ofY0 is a quadratic point ofH.

With a quadratic z H we associate its discriminant (z), namely (z) =b2 4ac whereaZ2 + bZ+ c is the minimal polynomial for z overZ with a >0.

Definition 3.8. For a special subvarietyY as above we define the complexity

(Y) = max

(z), N(g)

over all the coordinates z of Y0 and allg GL+2(Q) involved in the definition of

theYi, i 1.

Note that a weakly special subvariety has a certain number of non-special con-ditions, namely the number of coordinates ofY0 which are not quadratic, and isspecial just if this number is zero.

Further, weakly special subvarieties come in families. Given a strict partitionR= (R0, R1, . . . , Rk) we may form a new strict partition Sin which the elementspreviously inR0 are made into individual parts, the partsR1, . . . , Rk are retained,butS0 is empty. Now a weakly special subvariety Wof typeR comes with a choiceof some elements gi GL

+2(Q) for the parts Ri. This same choice determines

a weakly special subvariety T of type S which is in fact special (even stronglyspecial, as there are no fixed coordinates). We call T the special envelope oftype R. The variety Wnow lies in the family of weakly special subvarieties ofTcorresponding to choices for the fixed coordinatesR0. It is thus a family of weaklyspecial subvarieties ofT parameterised by HR0 . We will call the members of thefamily translatesof the strongly special subvariety ofHR1...Rk corresponding tothe given elements gi, the space of translates being HR0 . The translate of T byt HR0 we denote Tt.


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We apply the same terminology to the images in Y(1)n. In particular, we havethe following.

Definition 3.9. A weakly special subvariety of Y(1)n is the image j(Y) whereY is a weakly special subvariety ofHn, and is special if Y is special (for someor equivalently all possible choices forY). The complexity of a special subvarietyT Y(1)n, denoted(T), is equal to the complexity ofY(any choice will give thesame complexity due to theSL2(Z) invariance).

As observed the weakly special subvarietyY Hn is a fibre in a family of weaklyspecial subvarieties of some special subvariety T. Thus, the image under j ofYand the other translates are algebraic subvarieties ofj (T).

4. Geodesic-optimal subvarieties

Throughout this section and if not further specified let Xbe a mixed Shimuravariety or a semi-abelian variety defined over C.

The collection of weakly special subvarieties, like the collection of special subvari-eties, is closed under taking irreducible components of intersection and contains theambient variety, so there is a smallest weakly special subvariety Ageo containingany subvariety A X. We denote by

geo(A) = dimAgeo dim A

thegeodesic defect ofA.

Definition 4.1. Let V X be a subvariety. A subvariety A V is said to begeodesic-optimal (for V in X) if there is no subvariety B withA B V suchthat

geo(B) geo(A).

As for the defect, the specification ofV andXwill generally be suppressed.

Note: what we call geodesic-optimal has been termed cd-maximal (co-dimension maximal) in the multiplicative context by Poizat [38]; see also[4].

Again, if A V is geodesic-optimal then it is a component of V Ageo.Further, as special subvarieties are weakly special, we have, for any V andA V,Ageo A and so geo(A) (A). In contrast to the defect, the geodesic defectof a singleton is always 0. Therefore, a singleton is geodesic-optimal in V if andonly if it is not contained in a coset of positive dimension contained in V .

Definition 4.2. We say that X has the defect condition if for any subvarietiesA B Xwe have

(B) geo(B) (A) geo(A).

Proposition 4.3. The defect condition holds

(i) ifX= Gnm is an algebraic torus,(ii) or ifX is an abelian variety,

(iii) or ifX= Y(1)n.

Proof. LetA B Xbe as in Definition4.2. For (i) letB Gnm and let

L= {(a1, . . . , an) Zn; xa11 x

ann is constant onB},

M={(a1, . . . , an) Zn; xa11 x

ann is constant and a root of unity onB}


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be free abelian groups. Then codimB = rank M and codimBgeo = rank L, sothat

(B) geo(B) = rank L/Mis the multiplicative rank of constant monomial functions on B. Such functionsremain constant and multiplicatively independent on A.

To prove (ii) let A and B be subvarieties ofX. The coset Bgeo is a translateof an abelian subvariety Y of X. Let us write : X X/Y for the quotientmorphism. We fix an auxiliary point P A(C).

We remark that P +Y is a torsion coset that contains P+Y. As P B(C)we also have B P+Y and thus B P +Y. We apply, which has kernelY, to find (B) (P) (A). We conclude

(4.1) dimB dimBgeo = dimB dim Y= dim (B) dim (A)

where the fact that B contains a translate ofYand basic dimension theory areused for the second equality.

The torsion coset A is the translate of an abelian subvariety Z of X by atorsion point. The fibres of |A : A (A) contain translates of Y Z.Using dimension theory we find dim (A) dimA dim Y Z. We observedim Y Z dimAgeo and so dim (A) dimA dimAgeo. Now let uscombine this bound with (4.1) to deduce

dimB dimBgeo dimA dimAgeo.

This inequality enables us to conclude

(B) = dimB dim B

dimA + dimBgeo dimAgeo dim B

=geo(B) geo(A) + (A),

as desired.In case (iii), (B) geo(B) is just the number of constant and non-special

coordinates ofBgeo. Then any weakly special subvariety containingA (which isnon-empty) but contained in Bgeo, and in particularAgeo, has also at least thatmany non-special constant coordinates.

Conjecture 4.4. Every mixed Shimura variety (and every weakly special subvari-ety) has the defect condition.

Proposition 4.5. Let X have the defect condition, e.g. X is an abelian varietyorY(1)n, and letV Xbe a subvariety. An optimal subvariety ofV is geodesic-optimal inV.

Proof. Let A V be an optimal subvariety and consider a subvariety B withA B V such that geo(B) geo(A). Then(B) geo(B) (A) geo(A)and so

(B) = geo(B) + (B) geo(B) geo(A) + (A) geo(A).

SinceA is assumed optimal we must have B = A, and so A is geodesic optimal.Finally, the proposition applies to abelian varieties and Y(1)n because of Propo-

sition4.3.


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5. Weak complex Ax

In this section we formulate various Ax-Schanuel type Conjectures. In the con-

text of abelian varieties, these conjectures will be theorems. But their modularcounterparts are largely conjectural.

As a warming-up let us recall a consequence of Ax-Schanuel [ 2]in the complexsetting.

Let now A= be an irreducible complex analytic subspace of some open U Cn

such that locally the coordinate functions z1, . . . , zn and exp(z1), . . . , exp(zn) aredefined and meromorphic on A.

Definition 5.1. The functionsz1, . . . , zn onA are called linearly independent overQ modC if there are no non-trivial relations of the form

qizi = c onA where

qi Q andc C.

The linear independence of the zi over Q mod C on A is equivalent to themultiplicative independence of the exp(zi) on A.

Theorem 5.2 (Ax). If the functionsz1, . . . , zn onA are linearly independent overQ modC then

trdeg CC(z1, . . . , zn, exp(z1), . . . , exp(zn)) n+ dim A.

5.1. The abelian setting. We will state 2 variations on Axs Theorem that aresufficient to treat unlikely intersections in abelian varieties.

LetXbe an abelian variety defined over C. We writeT0(X) for the tangent spaceof X at the origin. Moreover, there is a complex analytic group homomorphismexp : T0(X) X(C). Here we use the symbol exp instead of to emphasise thegroup structure.

Theorem 5.3 (Ax). LetUT0(X) be a complex vector subspace andz T0(X).LetK be an irreducible analytic subset of an open neighborhood ofz inz+ U. IfB is the Zariski closure ofexp(K) inX, thenB is irreducible and

geo(B) dim U dim K.

Proof. See Corollary 1[3].

The following statement is sometimes called the Ax-Lindemann-Weierstrass The-orem for abelian varieties. Theorem5.3 will be used in Section 6.1 whereas Ax-Lindemann-Weierstrass makes its appearance near the end in Section 9where weapply it in connection with a variant of the Pila-Wilkie Theorem. Ax-Lindemann-Weierstrass is sufficient to prove Theorem1.1,but Theorem1.3, situated in higherdimension, requires both statements. The reason seems to be that certain technicaldifficulties disappear in low dimension.

We may also consider T0(X) as a real vector space of dimension 2 dimX. Af-

ter fixing an isomorphism T0(X) = R2 dimX it makes sense to speak about semi-algebraic maps [0, 1] T0(X).

Theorem 5.4 (Ax). Let : [0, 1] T0(X) be real semi-algebraic and continuouswith|(0,1) real analytic. The Zariski closure inXof the image ofexp is a coset.

Proof. Clearly, we may assume that is non-constant. The Zariski closure B ofthe image exp(([0, 1])) is irreducible since exp is continuous and real analyticon (0, 1).


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By considering Taylor expansions around points of (0, 1), the restriction |(0,1)extends to a holomorphic map with target T0(X) and defined on a domainin C which contains (0, 1). By analyticity the image of exp lies in B andtrdeg CC(exp ) dim B.

As is real algebraic on [0, 1] we find trdeg CC() 1 and therefore,

trdegCC(, exp ) trdeg CC() + trdeg CC(exp ) 1 + dim B.

Let us apply the one variable case of Axs Theorem 3 [ 3] to the holomorphicfunction t (t+ 1/2) (1/2) defined in a neighborhood of 0 C. Accordingto the inequality of transcendence degrees, the smallest abelian subvariety HXcontaining all values exp((t+ 1/2) (1/2)) as t runs over U has dimension atmost dim B. Therefore, B = exp((1/2)) + Hwhich is what we claimed.

5.2. A product of modular curves. Now we suppose that X= Y(1)n and that: Hn X(C) is the n-fold cartesian product of thej -function.

Let again A = be an irreducible complex analytic subspace of some open

U Hn, so that locally the coordinate functionsz1, . . . , znand j (z1), . . . , j(zn) aredefined and meromorphic on A, and we have a finite set {Dj} of derivations withrank(Djzi) = dim A, the rank being over the field of meromorphic functions on A.

Definition 5.5. The functionsz1, . . . , zn onA are called geodesically independentif nozi is constant and there are no relationszi= gzj wherei =j andg GL

+2(Q).

The geodesic independence of the zi is equivalent to the j(zi) being modularindependent, i.e. non-constant and no relations N(j(zk), j(z)) wherek = andN(X, Y) is a modular polynomial.

The following conjecture might be considered the analogue of Ax-Schanuel forthej -function in a complex setting.

Conjecture 5.6 (Complex Modular Ax-Schanuel). In the above setting, if the

zi are geodesically independent thentrdegCC(z1, . . . , zn, j(z1), . . . , j(zn)) n + dim A.

It evidently implies a weaker two-sorted version that, with the same hypothe-ses, we have the weaker conclusion

trdeg CC(z1, . . . , zn) + trdeg CC(j(z1), . . . , j(zn)) n+ dim A.

This conjecture is open beyond some special cases (Ax-Lindemann[32] and Mod-ular Ax-Logarithms[21]).

We pursue now some more geometric formulations. To frame these we need somedefinitions.

Definition 5.7. By a subvariety ofU Hn we mean an irreducible component (inthe complex analytic sense) ofW U for some algebraic subvarietyW Cn.

Definition 5.8. A subvariety W Hn is called geodesic if it is defined by somenumber of equations of the forms

zi = ci, ci C; zk = gkz, g GL+2(Q).

These are the weakly special subvarieties in the Shimura sense.

Definition 5.9. By a component we mean a complex-analytically irreducible com-ponent ofW 1(V) whereW U andV Xare algebraic subvarieties.


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LetA be a component ofW 1(V). We can consider the coordinate functionsziand their exponentials as elements of the field of meromorphic functions (at leastlocally) on A, and we can endow this field with dim A derivations in such a waythat rank(Dzi) = dim A. Then (with Zcl indicating the Zariski closure)

dim W dimZcl(A) = trdeg CC(z1, . . . , zn),

dim V dim Zcl((A)) = trdeg CC(j z1, . . . , j zn)

and the two-sorted Modular Ax-Schanuel conclusion becomes

dim W+ dim V dim X+ dim A

provided that the functions zi are geodesically independent.

This condition is equivalent toAbeing contained in a proper geodesic subvariety.Let us takeU to be the smallest geodesic subvariety containing A. Let X = exp U,which is an algebraic subtorus ofX, and put W =W U, V =V X. We canchoose coordinateszi, i= 1, . . . , dim A which are linearly independent over Q mod

C and derivations such that rank(Dzi) = dim A. We get the following variant ofAx-Schanuel in this setting.

Conjecture 5.10 (Weak Complex Ax (WCA): Formulation A.). LetU be a geo-desic subvariety ofU. PutX = exp U and letA be a component ofW 1(V),whereWU andV X are algebraic subvarieties. IfA is not contained in anyproper geodesic subvariety ofU then

dim A dim V+ dim W dim X.

I.e. (and as observed still more generally by Ax [3]), the intersections ofW and1(V) never have atypically large dimension, except when A is contained in aproper geodesic subvariety. It is convenient to give an equivalent formulation.

Definition 5.11. Fix a subvarietyV X.

(i) A component with respect to V is a component of W1(V) for somealgebraic subvarietyWU.

(ii) IfA is a component we define its defect by(A) = dim Zcl(A) dim A.(iii) A componentA w.r.t. V is called optimal forV if there is no strictly larger

componentB w.r.t. V with(B) (A).(iv) A component A w.r.t. V is called geodesic if it is a component of W

1(V) for some weakly special subvarietyW= Zcl(A).

Conjecture 5.12(WCA: Formulation B.). LetV Xbe a subvariety. An optimalcomponent with respect to V is geodesic.

Formulation B is the statement we need. However, the two formulations areequivalent, and the proof of their equivalence is purely formal and applies in thesemiabelain setting and indeed quite generally.

Proof that formulation A implies formulation B. We assume Formulation A and sup-pose that the componentA ofW 1(V) is optimal, whereW= Zcl(A). SupposethatU is the smallest geodesic subvariety containing A, and letX =(U). ThenW U. Let V = V X. ThenA is optimal for V in U, otherwise it wouldfail to be optimal for V in U. Since A is not contained in any proper geodesicsubvariety ofU we must have

dim A dim W+ dim V dim X.


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LetB be the component of1(V) containingA. ThenB is also not contained inany proper geodesic subvariety ofU, so, by Formulation A,

dim B dim V

+ dimZcl(B) dim X

.But dim B = dim V, whence dim Zcl(B) = dim X, and so Zcl(B) =X, and B isa geodesic component. Now

(A) = dim W dim A dim X dim V =(B)

whence, by optimality, A = B.

Proof that formulation B implies formulation A. We assume Formulation B. LetU be a geodesic subvariety ofU, put X = (U). SupposeV X, W U arealgebraic subvarieties and A is a component ofW 1(V) not contained in anyproper geodesic subvariety ofU. There is some optimal component B containingA, and B is geodesic, but sinceA is not contained in any proper geodesic,B mustbe a component of1(V) with Zcl(B) = U and we have

dim W dim A (A) (B) = dim X dim V

which rearranges to what we want.

As already remarked, WCA holds for (semi-)abelian varieties, by Ax [3] (see also[23]).

To conclude we note that a true Modular Ax-Schanuel should take into accountthe derivatives ofj . A well-known theorem of Mahler [26] implies that j, j, j arealgebraically independent over C as functions on H, and it is well-known too that

j Q(j, j, j) (see e.g. [5]).

Conjecture 5.13(Modular Ax-Schanuel with derivatives). In the setting of Mod-ular Ax-Schanuel above, ifzi are geodesically independent then

trdegCC(zi, j(zi), j(zi), j

(zi)) 3n + dim A.

6. A finiteness result for geodesic-optimal subvarieties

6.1. The abelian case. Suppose Xis an abelian variety defined over C. In thissection we prove the following finiteness statement on geodesic-optimal subvarietiesof a fixed subvariety ofX.

Proposition 6.1. For any subvariety V X there exists a finite set of abeliansubvarieties ofX with the following property. IfA is a geodesic-optimal subvarietyofV, thenAgeo is a translate of a member of the said set.

Any positive dimensional, geodesic-optimal subvariety A V is -anormalmaximal for a certain in Remonds terminology[43]. His Lemme 2.6 and Propo-sition 3.2 together imply thatAgeo is a translate of an abelian subvariety coming

from a finite set that depends only on V. Thus our proof Proposition 6.1 canbe regarded as an alternative approach to Remonds result using the language ofo-minimal structures.

We retain the meaning of the symbol T0(X) from Section 3 and write exp :T0(X) X(C) for the exponential map. It is a holomorphic group homomorphismbetween complex manifolds whose kernel is the period lattice of X. We choosea basis of the period lattice and identify T0(X) with R2g as a real vector space.However, we continue to use both symbols T0(X) and R

2g; the former is useful to


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emphasise the complex structure and the latter is required as an ambient set for ano-minimal structure.

The open, semi-algebraic set (1, 1)2g contains a fundamental domain for theperiod latticeZ2g R2g in real coordinates. Under the identification R2g =T0(X)we fixed above, we may consider (1, 1)2g as a domain in T0(X).

LetV be a subvariety ofX. Then

V= exp |1(1,1)2g(V(C))

is a subset ofR2g and definable inRan. But under the isomorphism mentioned be-fore,Vis also a complex analytic subset of (1, 1)2g T0(X). Thus it is a complexanalytic space. The interplay of these two points of view will have many conse-quences. For an in-depth comparision between complex and o-minimal geometrywe refer to Peterzil and Starchenkos paper [30].

Indeed, suppose Z is an analytic subset of a domain in T0(X) and z Z.Then some open neighborhood ofz in Zis definable in Ran as Zis defined by the

vanishing of certain holomorphic functions. So the dimension dimzZ ofZ at z asa set definable in Ran is well-defined. But there is also the notion of the dimensionofZat z as a complex analytic space [19].

In this section we will add the subscript C to the dimension symbol to signifythe dimension as a complex analytic space.

As dimC = 2 = 2 dimCC the following lemma not surprising.

Lemma 6.2. LetZbe an analytic subset of a finite dimensionC-vector space. Ifz Z thendimzZ= 2 dimC,zZ.

Proof. Locally atz the complex analytic space Zis a finite union of prime compo-nents, each of which is analytic in a neighborhood ofz in Z. Without loss of gener-ality we may assume that Zis irreducible at z . After shrinking Z further we may

suppose thatZis irreducible, definable in Ran, and satisfies dimzZ= dim Z. We fixa decomposition ofZinto cells D1 DNand writeZ =Z

dimDi


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Then Fr is definable in Ran by standard properties of o-minimal structures, forexample by Proposition 1.5, chapter 4[15]taking local dimensions is definable. Weset

Er =

(z, M) Fr; for all M O with ker M ker M :

dimz V (z+ ker M)< dimz V (z+ ker M)

which is also definable in Ran.

Axs Theorem5.3 for abelian varieties will be used in

Lemma 6.3. (i) If (z, M) Er there is an abelian subvariety Y X withT0(Y) = ker M.

(ii) The set{ker M; (z, M) Er}

is finite.(iii) LetA be a geodesic-optimal subvariety ofV and letAgeo be a translate of

an abelian subvarietyY. If r = dimC Y andM O withT0(Y) = ker Mthen(z, M) Er for somez (1, 1)2g.

Proof. Say (z, M) Er is as in (i). We apply Axs Theorem5.3to U= ker M. Forthis we fix a prime component (in the complex analytic sense) K V (z + U)with dimC,zK= dimC,z V (z+U). By shrinkingKto an open neighborhood ofz we may assume that Kis irreducible and definable in Ran. Let B Xbe as inAxs theorem.

IfBgeo is a translate of the abelian subvariety Y XthenK z + T0(Y) andso

K z + U T0(Y).

Observe thatU T0(Y) is the kernel of an element inO by property (ii) above. Bythe definition ofEr we must have U T0(Y) = U. Therefore, UT0(Y).

Next we prove that equality holds. Indeed, ifU T0(Y), then we may test U

against any N O with ker N=T0(Y) in the definition ofFr. Therefore,

dim U dimzK= dim U dimz V (z+ U)


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LetA and Ybe as in (iii). Since A is a geodesic-optimal subvariety ofV it mustbe an irreducible component ofV Ageo. Let us fix z Vsuch that exp(z) is asmooth complex point ofAthat is not contained in any other irreducible componentofV Ageo.

We first prove (z, M) Fr by contradiction. So suppose there exists N Owith T0(Y) ker N and

(6.1) dim T0(Y) dimz V (z+ T0(Y)) dim ker N dimz V (z+ ker N).

As in (i) we fix a prime component K ofV (z+ ker N) that passes through zwith dimC,zK = dimC,z V (z+ ker N) and is irreducible. LetB be the Zariskiclosure of exp(K), then Axs Theorem impliesgeo(B) dimCker N dimC K. Asexp is locally biholomorphic our choice ofz implies thatz is a smooth point of thecomplex analytic setV (z + T0(Y)) which has dimension dimC Aat this point. So

(6.2) geo(A) = dimC T0(Y) dimC A dimCker N dimC Kgeo(B)

follows from (6.1) after dividing by 2.By smoothness, the intersectionV (z+ T0(Y)) has a unique prime component

K passing through z. The dimension inequality for intersections, cf. Chapter 5,3 [19], implies

dimC,zK (z+ T0(Y)) dimC,zK+ dimC T0(Y) dimCker N.

Inequality (6.1) and the discussion above imply that the right-hand side is at leastdimC A. But K(z+ T0(Y)) V (z+ T0(Y)) and on comparing dimensionsat z we find thatK (z+ T0(Y)), and a fortiori K, contains a neighborhood ofzin K. This implies A B. But (6.2) and the fact that A is a geodesic-optimalsubvariety forces A = B. So dimC K dimC A and (6.2) applied again yieldsdimC T0(Y) dimCker N, a contradiction.

Second, we will show (z, M) Er. Suppose on the contrary that there isM Owith ker M T0(Y) and

dimz V (z+ ker M) = dimz V (z+ T0(Y)).

The set on the right is a complex analytic space, smooth at z, and contains theformer. So V (z +ker M) and V (z +T0(Y)) coincide on an open neighborhood ofz in (1, 1)2g. Therefore, an open neighborhood of 0 inA(C) exp(z) is containedin the group exp(ker M); here and below we use the Euclidean topology. Saidgroup need not be algebraic or even closed, but it does contain an open, non-emptysubset of the complex points of

(A exp(z)) + + (A exp(z))

dimX terms

.

This sum equals A exp(z)= Y= exp(T0(Y)). Hence T0(Y) ker M, which isthe desired contradiction.

Proof of Proposition6.1. We will work with O = End(T0(X)). Suppose that Ais a geodesic-optimal subvariety ofV. Let us fix M O such that Ageo is thetranslate of an abelian subvariety whose tangent space is ker M. Then ker M liesin a finite set by Lemma 6.3 parts (ii) and (iii). So Ageo is the translate of anabelian subvariety ofXcoming from a finite set.
http://-/?-http://-/?-


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LetV Xbe a subvariety. We brief discussion the connection between Propo-sition6.1and anomalous subvarieties as introduced by Bombieri, Masser, and Zan-nier[10].

Definition 6.4. LetXandVbe as above. A subvarietyA Vis called anomalousif

(6.3) dim A max{1, dimAgeo+ dim V dim X+ 1}.

If in addition A is not contained in any strictly larger anomalous subvariety ofV, then we callA maximal anomalous. The complement in Vof the union of allanomalous subvarieties ofV is denoted byVoa.

Any maximal anomalous subvariety A ofV is geodesic-optimal. Indeed, afterenlargening there is a geodesic-optimal subvariety BofV withA B and geo(B)geo(A). So

dim B dimBgeo dimAgeo+ dim A dimBgeo+ dim V dim X+ 1due to (6.3). Since dim B dim A 1 we see that B is anomalous. As A ismaximal anomalous we findB = A. SoA is geodesic-optimal.

According to Proposition 6.1, Ageo is the translate of an abelian subvarietycoming from a finite set which depends only on V. Let Y be such an abeliansubvariety. By a basic result in dimension theory of algebraic varieties the set ofpoints V(K) at which X X/Y restricted to Vhas a fibre greater or equal tosome prescribed value is Zariski closed in V . It follows that Voa is Zariski open inV; we may thus use the notation Voa(K) for K-rational points inVoa.

Openness ofVoa previously known due to work of Remond [43]and proved earlierin the toric setting by Bombieri, Masser, and Zannier [10].

We remark that Voa is possibly empty. Moreover,Voa = if and only if thereexists aYas above such that

dim (V)< min{dim X/Y, dim V}

where : XX/Y.

6.2. Mobius varieties. LetX=Y(1)n.It is convenient to introduce a family subvarieties ofHn parameterised by choices

of elements ofH and GL+2(R) in which weakly special subvarieties are the fibres cor-responding to the GL+2(R) parameters having their image under scaling in SL2(R)lying in the image of GL+2(Q). Observe that this image is a countable set. In[32]these were termed linear subvarieties but the denotation Mobius seems to bemore appropriate.

We takezias coordinates in Hn and gi, i= 2, . . . , nas coordinates on GL+2(R)

n1

The family of Mobius curves in Hn

is the locusM{1,...,n} Hn GL+2(R)

n1

defined by the equations zi= giz1, i= 2, . . . , n. We view this as a family of curvesin Hn parameterised byg = (g2, . . . , gn) GL

+2(R)

n1. For a subsetR {1, . . . , n}we defineMR to be the family of Mobius curves on the product of factors ofHn overindices inR, parameterised by the corresponding factors of GL+2(R)

n excluding thesmallest one which plays the role ofz1, which we will denote GL

+2(R)

Ri .


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LetR = (R0, R1, . . . , Rk) be a strict partition as near Definition3.6. Define thefamily of Mobius subvarieties of type R to be the locus

MR Hn HR0 k

i=1

GL+2(R)Ri

defined by equations placing the HRi GL+2(R)Ri point in MRi for i = 1, . . . , k,

and eachR0-coordinate inHn is set equal to the corresponding coordinate inHR0 .

For each choice of parameters

t MR= HR0

ki=1

GL+2(R)Ri

the corresponding fibre MRt is aMobius subvarietyHn.

Like weakly special subvarieties, Mobius subvarieties come in families of trans-

lates. For a fixed g ki=1GL

+2(R)

Ri , the choices ofz HR0 give a family of

Mobius subvarieties, the translates of the corresponding strongly Mobius sub-varietyMg of the appropriate subproduct ofHn, and the totality of the translatesform a Mobius subvariety with no fixed coordinates.

A component A Hn is contained in some smallest Mobius subvariety LA, hasa Mobius defect

M(A) = dim LA dim A

A componentA j1(V) will be called Mobius optimal(forV in X) if there is nocomponentB with A B j1(V) and M(B) M(A).

Proposition 6.5. Assume WCA. LetV Xbe a subvariety. Then the set of

gk

i=1

GL+2(R)Ri

such that some translate ofMg intersectsj1(V) in a component which is Mobiusoptimal forVis finite modulo the action by

iSL2(Z)

Ri .

Proof. The condition is

iSL2(Z)Ri invariant, so the assertion is that suchg come

in finitely many

iSL2(Z)Ri orbits. By WCA, any such g corresponds to a weakly

special subvariety, and so the g in question belong to a countable set. However,every such g has a translate under

iSL2(Z)

Ri for which the optimal componenthas points of its full dimension in some fixed fundamental domain, say Fn0 , and therethe condition of optimality may be checked definably by considering dimensions ofthe intersection of1(V) Fn0 with Mobius subvarieties over the whole space ofthem, which is definable. Thus, there is a definable (in Ran,exp) countable andhence finite set ofg which contains a representative of every

iSL2(Z)

Ri orbit of

such g .

As observed, the g above all correspond to weakly special families; however,everyg corresponding to a weakly special family having a translate with a geodesic-optimal intersection will also appear in this set, as such g(by WCA) are in particularoptimal among Mobius varieties. We conclude a modular version of Proposition 6.1.

Proposition 6.6. Assume WMA. Let V X be a subvariety. Then there isa finite set of basic special subvarieties such that every weakly special subvariety


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which has a geodesic-optimal component in its intersection withVis a translate ofone of these.

Finally, we observe that special subvarieties of type R arise as fibres ofMR ofpoints ofMR with suitable rationality properties. Specifically, the coordinates inGL+2(R)

Ri are rational and those in HR0 are quadratic; let us call these specialpoints. Of course the same fibre arises from non-special choices of the parametertoo.

Proposition 6.7. There is an absolute constantc >0 with the following property.Let T Y(1)n be a special subvariety with complexity (T) containing a pointPY(1)n with pre-imageQ Fn0 . Then there exists a special pointt MR with

H(t) c(T)10

such thatMRt is a component of the pre-image ofT andQ MRt .

Proof. This follows from Lemmas 5.2 and 5.3 of [21].

7. Counting semi-rational points

In this section we will work in a fixed o-minimal structure over R. Our goalis to count points on a definable set where a certain coordinates are algebraic ofbounded height and degree and the rest are unrestricted. We will use our result tostudy unlikely intersections in abelian varieties.

Let us first fix some notation. Letk 1 be an integer. We define the k-heightof a real number y R as

Hk(y) = min

max{|a0|, . . . , |ak|}; a0, . . . , ak coprime integers, not all zero,

witha0yk + + ak = 0

using the convention min = +. A real number has finitek-height if and only if

it has degree at mostk over Q. Letm 0 be an integer. Fory = (y1, . . . , ym) Rmwe set

Hk(y) = max{Hk(y1), . . . , H k(ym)}

and abbreviateH(y) =H1(y) ify Rm. IfZ Rm is any subset, we define

Z(k, T) = {y Z; Hk(y) T}

forT 1.Ifis a mapping between two sets, then () will denote the graph of. Suppose

n 0 is an integer. If we state that a subset F Rm Rn is a family parametrisedbyRm, thenFy stands for the projection of{y} Rn F to Rn ify Rm.

LetZ Rm Rn be a family parametrised by Rm. We want to determine thedistribution of points (y, z) Zwherey has k-height at mostTwithout restrictingz. For a real number T 1, we define

Z(k, T) = {(y, z) Z; Hk(y) T} .

For technical reasons it is sometimes more convenient to work with

Z,iso(k, T) = {(y, z) Z(k, T); z is isolated in Zy} .

Let l 0 be an integer. We will use the second-named authors generalisation,stated below, of the Pila-Wilkie Theorem [34] to prove the following result fordefinable families.


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Theorem 7.1. LetF Rl Rm Rn be a definable family parametrised byRl

and > 0. There is a finite numberJ=J(F ,k,) of block families

W(j)

Rkj

Rl

Rm

, j {1, . . . , J }parametrised byRkj Rl, for each suchj a continuous, definable function

(j) :W(j) Rn,

and a constantc = c(F ,k,) with the following properties.

(i) For allj {1, . . . , J } and all(t, x) Rkj Rl we have

((j))(t,x) {(y, z) Fx; z is isolated inF(x,y)}.

(ii) Say x Rl and Z = Fx. If T 1 the set Z,iso(k, T) is contained in

the union of at mostcT graphs((j))(t,x) for suitablej {1, . . . , J } and

t Rkj .

What follows is a useful corollary of the result above. Its assertion deals with

Z

(k, T) and not Z,iso

(k, T) which appears in the theorem.Corollary 7.2. LetF and be as in Theorem7.1. We let1 and2 denote theprojections Rm Rn Rm and Rm Rn Rn, respectively. There exists aconstantc = c(F ,k,)> 0 with the following property. Sayx Rl and letZ=Fx.IfT 1 and Z(k, T) with

(7.1) #2()> cT,

there exists a continuous and definable function: [0, 1] Zsuch that the follow-ing properties hold.

(i) The composition 1 : [0, 1] Rm is semi-algebraic and its restrictionto (0, 1) is real analytic.

(ii) The composition2 : [0, 1] Rn is non-constant.(iii) We have2((0)) 2().

(iv) If the o-minimal structure admits analytic cell decomposition, then|(0,1)is real analytic.

Here is the second-named authors counting theorem involving blocks.

Theorem 7.3(Theorem 3.6[32]). LetF RlRm be a definable family parametrisedbyRl and > 0. There is a finite numberJ=J(F, ) of block families

W(j) Rkj Rl Rm, j = 1, . . . , J ,

each parametrised byRkj Rl, and a constant c = c(F ,k,) with the followingproperties.

(i) For all(t, x) Rkj Rl and allj {1, . . . , J } we haveW(t,x) Fx.

(ii) For allx Rl andT 1 the setFx(k, T) is contained in the union of at

mostcT blocks of the formW(j)

(t,x) for suitablej = 1, . . . , J andt Rkj .

Proof of Theorem7.1. We refer to van den Driess treatment of cells in Chapter3 [15]. His convention for a cell C Rm Rn has the following advantage whenconsidering it as a family parametrised by Rm. Ify Rm then Cy Rn is eitherempty or a cell of dimension dim C 1.

We begin the proof of the theorem with a reduction step. Let us consider theset

F ={(x,y,z) F; z is isolated in F(x,y)}.


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We claim that it is definable. Indeed, let C1 CNbe a cell decomposition ofF. ThenF(x,y) = (C1)(x,y) (CN)(x,y) and each (Ci)(x,y) is either empty ora cell. The dimension of a non-empty (C

i)

(x,y)does not depend on (x, y) and is

the same locally at all points. Therefore,F is a union of a subclass of theCi andhence definable.

SinceF is definable it suffices to complete the proof withFreplaced byF. Wethus assume that z is isolated in F(x,y) for all (x,y,z) F.

By general properties of an o-minimal structure, the number of connected com-ponents in a definable family is finite and bounded from above uniformly. So#F(x,y) c1 for all (x, y) R

l Rm wherec1 is independent ofx and y .

Let : RlRmRn RlRm denote the natural projection. ThenE(1) =(F)is a definable set. By Definable Choice, Chapter 6 Proposition 1.2(i)[15], there isa definable function f(1) : E(1) Rn whose graph (f(1)) lies in F. This graphis definable and so is F (f(1)). But the cardinality of a fibre of F (f(1))considered as a family over Rl Rm is at most c1 1.

IfF = (f1), Definable Choice yields a definable function f(2) : E(2) Rn onE(2) = (F (f1)) whose graph is inside F (f1). The fibre above (x, y) ofF ((f1) (f2)) has at mostc1 2 elements.

This process exhausts all fibres ofFafterc2 c1steps. We get definable familiesE(1), . . . , E (c2) Rl Rm parametrised by Rl and definable functions

(7.2) f(i) :E(i) Rn for i {1, . . . , c2} withc2i=1

(f(i)) = F.

We can decompose eachE(i) into finitely many cells on which f(i) is continuous.So after possibly increasing c2 we may suppose that each f(i) is continuous anddefinable.

Ifx Rl, not all coordinates of a point in Fx (k, T) need to be algebraic. But

the firstm coordinates are and lead to points ofk-height at most Ton someE(i)x .These points can be treated using Theorem7.3applied to the familyE(i). For everyi {1, . . . , c2} we obtain Ji block families W(i,j) Rki,j Rl Rm parametrised

by Rki,j Rl wherej {1, . . . , J i}. They satisfyW(i,j)(t,x) E

(i)x for (t, x) Rki,j Rl

and account for all points ofk-height at mostT on E(i)x .

Note that if (t,x,y) W(i,j), then (x, y) E(i). We consider the function

(i,j) :W(i,j) Rn defined by (t,x,y)f(i)(x, y).

It is definable, being the composition of two definable functions: a projection andf(i). Moreover, (i,j) is continuous by our choice of theE(i). Observe ((i,j))(t,x)

Fx for all (t, x) Rki,j Rl and this will yield (i).

Suppose x Rl and (y, z) Z(k, T) with Z = Fx. The point (x,y,z) Flies on the graph of some f(i) by (7.2). Hence z = f(i)(x, y) with (x, y) E(i).

By definition we have y E(i)x . So y E

(i)x (k, T) since y has k-height at most

T. Supposec(E(i), k , ) is the constant from Theorem 7.3. Theny is inside some

W(i,j)(t,x)

wherej and t are allowed to vary overc(E(i), k , )T possibilities. Therefore,

(t,x,y) W(i,j) and (t ,x,y,z) ((i,j)) or equivalently, (y, z) ((i,j))(t,x).

Part (ii) and the theorem follow after renumbering the (i,j) andW(i,j).


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Proof of Corollary7.2. The constantcfrom the this corollary comes from Theorem7.1applied to F,k, and.

Letx Rl andZ=Fx

Rm Rn. SupposeT1 satisfies

(7.3) #2()> cT

with Z(k, T) as in the hypothesis.First, let us suppose Z,iso(k, T). We fix any (y, z) Z,iso(k, T); then

Hk(y) Tand the connected component ofZy containingzhas positive dimension.This component is definably connected. So we may fix a definable and continuouspath : [0, 1] Zy connecting (0) = z with any other point (1) = z of saidcomponent. Properties (i)-(iii) follows with the function(t) = (y, (t)) Z fort [0, 1]. Rescaling implies (iv).

We have reduced to the case Z,iso(k, T). By our Theorem7.1 the set iscontained in the union of at mostcT graphs of continuous and definable functions.The Pigeonhole Principle and (7.3) yield two (y, z), (y, z) on the same graph

with(7.4) z = 2(y, z) =2(y

, z) =z .

So there is a block family W Rk Rl Rm and a continuous, definable function : W Rn with (y, z), (y, z) ()(t,x) for a certain t R

k. Moreover,()(t,x) Z.

The fibreW(t,x)is a block containingyandy. A block is connected by definition.

As above this means that there is a continuous, definable function : [0, 1] W(t,x)with (0) =y and (1) =y . ButW(t,x), being a block, is locally a semi-algebraicset. That is, for any s [0, 1] the point (s) has a semi-algebraic neighborhood inW(t,x). Because [0, 1] is compact we may assume that is semi-algebraic.

We set

(s) = ((s), (t,x,(s)))

and this yields a function : [0, 1] Zwhich we show to satisfy all points in theassertion.

The function is continuous and definable as and possess these properties.We note that 1 = is semi-algebraic by construction. This yields the first

statement in (i).We also note (t,x,y) =z , so (0) = (y, z) and (iii) follows.We find (1) = (y, z) in a analog manner. Therefore, (7.4) implies 2((0))=

2((1)) and (ii) follows from this.To complete the proof of (i) we use the fact that Ralg admits analytic cell

decomposition. There exist 0 = a0 < a1 < < ak+1 = 1 such that each1 |(ai,ai+1) : (ai, ai+1) R

m Rn is real analytic. By continuity and (ii)the restriction of2 to some interval (ai, ai+1) is non-constant. Ifi is minimal

with this property, then 2((ai)) = 2((0)) = z . This will preserve (iii) after alinear reparametrisation of (ai, ai+1) to (0, 1). Thus we may suppose that1 |(0,1)is real analytic and this completes (i).

To prove (iv) we must assume that the ambient o-minimal structure admitsanalytic cell decomposition. As before we cover [0, 1] by finitely many open intervalsand points, such that 2 restricted to each open interval is real analytic. Weagain linearly rescale the first open interval on which 2 is non-constant to (0, 1).So (0,1) : (0, 1) R

m Rn is real analytic.


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SayZ Rm is a definable. The corollary applied (with m = n) to the diagonalembedding F = {(z, z); z Z} Rm Rm recovers Pila and Wilkies Theorem1.8[34].

8. Large Galois orbits

Let X be Y(1)n or an abelian variety defined over a field Kwhich we take tobe finitely generated over Q. Recall that we have a notion of complexity of specialsubvarieties of X, cf. Sections 3.1 and 3.2 for the abelian and modular cases,respectively. Suppose V X is a subvariety defined over K. We consider variousassertions.

(GO1) ForV X there exist c, >0 such that, for all P V,

[K(P) : K] c(P).

(GO2) As in (GO1) but only when the component{P}ofV P is optimal.(GO3) ForV Xthere existc, >0 such that, for all components WofV W,

[K(W) :K] c(W).

Now GO3 is simply too strong: e.g. ifV = X = Y(1)n then a component ofV T is T and T = T for any special subvariety, but some specials have largecomplexity and small Galois orbits (e.g. any strongly special TY(1)n).

Also GO1 seems to be very strong. TakingV = X = Y(1)n and P Qn, theexistence ofc, means that (P) is bounded as Pruns over all rational points,so that only finitely many P arise. But this could be true. E.g. in Y(1)2 weknow that there are only finitely many modular curves with non-CM, non-cuspidalrational points (Mazur). It is however stronger than we need.

ForV =Y(1)n we see thatPis a component if and only if it is special, and so thestatement of GO2 in that case reverts to the conjecture on Galois orbits of specialpoints. Also GO1 is odd ifPis transcendental, while in GO2 it must be algebraic

over K. For positive dimensional components we must depend on o-minimality(and WCA) to bring us to finitely many families, and reduce to considering theirtranslates.

Definition 8.1. LetKbe a field that is finitely generated overQ. SupposeX isY(1)n or an abelian variety defined overKandV X is a subvariety also definedover K. Let s 0. We say that LGOs(V) is satisfied if there exists a constant > 0 with the following property. For anyP V(K) such that{P} is an optimalsingleton ofV withdimP s we have

(8.1) (P) (2[K(P) : K]).

Ifr 0, we say thatX satisfiesLGOrs , ifLGOs(V) is satisfied for allV Xdefined overKabove withdim V r.

Finally, we say thatXsatisfiesLGO if it satisfiesLGOrs for allr, s 0.Conjecture 8.2. LetK be finitely generated overQ. If X is an abelian varietydefined overKor ifX= Y(1)n, thenXsatisfiesLGO.

One might expect an analog conjecture to hold in any mixed Shimura variety(or perhaps even any weakly special subvariety thereof).

Suppose K is a number field. For special points of Shimura varieties the bestresults known are those of Tsimerman [45]: lower bounds of the above form for


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the size of the Galois orbit for special points in the coarse moduli space of Abelianvarieties of dimension g, Ag, for g 6 (or on GRH for all g; see also [47]). Forunlikely intersections of curves with special subvarieties ofY(1)n partial results areobtained in [21].

To prove a uniform version (which one could frame as a ZP for Y(1)n Ck wherethe second factor is viewed with rational structure, as done in [ 33]), one would wantthat, for a family of subvarieties{Vt}, i.e. the fibres of someV Y(1)n Ck, theconstant for Vt in the family depends only on [Q(Vt) : Q], i.e. on the degree ofthe parametert overQ.

9. Unlikely intersections in abelian varieties

9.1. The arithmetic complexity of a torsion coset. In this section we willprove an upper bound on the arithmetic complexity of a torsion coset as introducedin Definition3.4.

Suppose X is an abelian variety of dimension g defined over a number field K

andL is an ample line bundle on X. To simplify notation we will suppose that Kis a subfield ofC. For example, this enables us to consider the tangent space T0(X)as aC-vector space. We letKdenote the algebraic closure ofK in C.

After replacingL by L L(1) we may assume that L is symmetric. Leth bethe Neron-Tate height on X(K) attached to L. We recall that the group of homo-morphisms between two abelian varieties is a finitely generated, free abelian group.Letd be the dimension of an abelian subvariety ofX. For the next proposition werequire

X(d) = sup{dim(X/H) rank Hom(X,X/H); HX is an abelian subvariety

overK with dim H= d}< +

where rank denotes the rank of a free abelian group and Hom(, ) the group ofhomomorphisms overK.

We observe that X(d) = 0 if and only ifd = dim X.

Proposition 9.1. There exists a constantc >0 depending onX andL such that

arith(P) c[K(P) : K]6g+1

anddegL Hc[K(P) : K]

60g4 max{1, h(P)}X(dimP)

for allP X(K) whereH=P P. In particular,

(P) c[K(P) : K]60g4

max{1, h(P)}X(dimP).

Bombieri, Masser, and Zannier [9] already employed essentially best-possibleheight lower bound due to Amoroso and David [1]to prove a weak form of the Zilber-Pink Conjecture for curves in Gnm. Later, Remond [41]developed this approach for

abelian varieties using the geometry of numbers. We will follow a similar line ofthought in this section. However, lower bounds of the same quality as Amoroso andDavids result are not known for a general abelian variety. One advantage of theo-minimal approach is that it can cope with a height lower bound as long as it ispolynomial in the reciprocal of the degree. Below, we will use such a lower bounddue to Masser [27]that applies to any abelian variety defined over a number field.For the sake of simplicity we state the estimates in a form that is weaker than whatMasser proved.


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Theorem 9.2 (Masser). There exists a constant c > 0 depending only on X,K,andL with the following property. SupposeP X(K) andD = [K(P) : K] withh(P)< c

1

D2g9

, thenPis a torsion point of order at mostcD6g+1

.Proof. This follows from the main theorem of [27] and the comments that followit.

GivenP X(K), a first consequence of Massers Theorem is a lower bound forthe [K(P) : K] in terms of the arithmetic complexity ofP. We do not yet needthe Neron-Tate height.

Lemma 9.3. There exists a constant c3 > 0 such that arith(P) c3[K(P) :K]6g+1 for allP X(K).

Proof. We note that the conclusion becomes stronger when replacingKby a fieldextension. So we may assume that all abelian subvarieties ofXare defined overK.

Bertrand proved that there is an integer c1 1 such that any abelian subvariety

Y Xhas a companion abelian subvariety Z X with Y +Z = X such thatY Z is finite and contains at most c1 elements. Ratazzi and Ullmo published aproof [40]of Bertrands Theorem. The point here is that c1 does not depend on Y.

Suppose thatPis a translate ofYby a torsion point. Let us write P=Q + Rwith Q Y(K) and R Z(K). A positive multiple ofP lies in Y(K). This and#Y Z < imply that R has finite order, say M. Massers Theorem impliesMc2[K(R) : K]6g+1.

By definition we have arith(P) M and thus arith(P) c2[K(R) :K]6g+1. It remains to bound [K(R) : K] from above in terms of [K(P) : K].

Suppose, Gal(K/K) with(R)=(R). If(P) =(P) thenP =Q + Ryields(Q) (Q) = (R) (R). This point is inY ZasY andZare definedoverK. This leaves at mostc1 possibilities for (R) (R). We conclude

[K(P) : K] = #{(P); Gal(K/K)}

1

c1#{(R); Gal(K/K)}

=[K(R) : K]

c1.

The lemma follows from the lower bound for [K(R) : K].

We setup some additional notation before we come to the proof of Proposition9.1.

SayY is a second abelian variety, also defined over K, equipped with an ampleand symmetric line bundle. Thus we have another Neron-Tate heighthY :Y(K)[0, ). We will assume that all elements in Hom(X, Y) are already defined overK. We set = rank Hom(X, Y). To avoid trivialities we shall assume 1, so in

particular dim Y 1. We set

Hom(X, Y)R = { Hom(X, Y) R; there is Hom(Y, X) R with = 1}

and fix a norm on Hom(X, Y) R. For example, we could take the norminduced by the Rosati involution coming from the two line bundles. We considerHom(X, Y) R with the induced topology. An element Hom(X, Y) R liesin Hom(X, Y)

R precisely when the linear map is surjective. Therefore,

Hom(X, Y)R

is an open, possibly empty, subset of Hom(X, Y) R.


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In this section, c1, c2, . . . denote positive constants that depend only on thesetwo abelian varieties, K, and the choosen line bundles.

The upper bound for the geometric part of the complexity involves the Neron-Tate height.

Lemma 9.4. Suppose Q > 1 and let P X(K) be in the kernel of a surjectiveelement ofHom(X, Y). There is a surjective Hom(X, Y) with

hY((P)) c4Q2/h(P) and c4Q.

Proof. By Lemmas 2 and 5[20] there is a compact subspaceK Hom(X, Y)R

,

Hom(X, Y), 0 K, and an integer q with 1 q Q such that hY((P))

c4Q2/h(P) andq0 c4Q1/. We emphasise thatc4 does not depend onP orQ.

The norm is bounded from above on the compact spaceK. So we obtain q0 + q0 c4Qafter increasing c4, if necessary.

We must show that is surjective. Recall that Hom(X, Y)R is open. Therefore,

there is Q0 1 such that Q > Q0 and 0 /q c4Q1/ imply /q Hom(X, Y)

R. In particular has a right inverse in Hom(Y, X) R. In this case it

must already have a right inverse in Hom(Y, X) Q by basic linear algebra. So is surjective.

Now ifQ Q0it suffices to take a fix surjective homomorphism for . By generalproperties of the Neron-Tate height hY((P)) is bounded from above linearly in

h(P), cf. expression (8) in Section 2[20]. The lemma follows after increasing c4 afinal time.

Lemma 9.5. Suppose P X(K) is in the kernel of a surjective element ofHom(X, Y) and D = [K(P) : K]. There exists a surjective Hom(X, Y) with(P) = 0 and

c6D6 dim Y+1+(2 dimY+9)/2 max{1, h(P)}/2.

Proof. Suppose Pis as in the hypothesis and let us abbreviate h = max{1, h(P)}.We suppose that is a surjective morphism with (P) = 0 and with minimalamong all such morphisms. Let c5 > 0 be the constant from Massers Theoremapplied toY. We will assume

(9.1) > 2(c4c5)1+/2D6 dim Y+1+(2 dimY+9)/2h/2,

with c4 from Lemma9.4, and derive a contradiction. This implies the propositionwith c6 = 2(c4c5)

1+/2.Let us define the integer

N= [c5D6dim Y+1].

Without loss of generality we havec5 1, so N1. We define further

Q=

2c4N.

Our assumption implies > 2c4c5D6 dim Y+1 2c4Nand so Q > 1. We apply

Lemma9.4 to find a surjective Hom(X, Y) with hY((P)) c4Q2/h and c4Q.


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Say 1 n N, then

n N c4N Q=

2 0 the rank of Hom(X, Y). We have dim H =

dim X dim Yby a dimension counting argument.Let X T0(X) denote the period lattice and tangent space ofXat the origin.We use the same norm on T0(X) as introduced in Section3.1. If Y T0(Y)denotes the period lattice of Y, then induces a linear map X Y. Sayg = dim H.

To proceed we apply an adequate version of Siegels Lemma to solve (i) = 0in linearly independent periods 1, . . . , 2g X with controlled norm. Indeed,we may refer to Corollary 2.9.9 [7], however the numerical constants there will notmatter for us. Siegels Lemma yields the first inequality in(9.2)

1 2g c72(gg) =c7

2dim Y c8D58g4 max{1, h(P)} dim Y,

the second one follows from the bound for we deduced further up and

(12 dim Y + 2 + (2dim Y + 9))dim Y (12g+ 2 + (2g+ 9)4g2)g 58g4

as 4g dim Y and dim Y g.Lemma3.1 and Hadamards Lemma yield

[ker : H]degL H= (dim H)![ker : H]vol(H) (dim H)!1 2g.

As [ker : H] 1 we get an upper bound for degL Hwhich yields the assertionwhen combined with (9.2).


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9.2. LGO and the Neron-Tate height. We begin by exhibiting a connectionbetween LGO, definition8.1,and height upper bounds on abelian varieties

Let Xbe an abelian variety defined over a number field Kand suppose L is asymmetric, ample line bundle onX. Lethdenote the associated Neron-Tate heightfunction. Observe that any optimal subvariety of a subvariety ofXdefined overKis defined overK.

Definition 9.6. LetV Xbe a subvariety defined overK. LetS 0. We defineOpt(V; S) to be the set of those A Opt(V) which contain a P A(K) with

h(P) (2[K(A) : K])S; here K(A) denotes the smallest field containingK overwhichA is defined.

In order to apply the counting strategy to study optimal subvarieties we mustfind a polynomial upper bound for the complexity of a torsion coset in terms of itsarithmetic degree. Since the inequality in Proposition9.1 also involves the heightwe make the following observation.

Proposition 9.7. LetV Xbe a subvariety defined overKand lets 0. ThenLGOs(V) is satisfied if there exist > 0 andS 0 such that

h(P) (2[K(P) : K])S(degLP)1

X(dimP)

for all optimal singletons {P} V with X(dimP) > 0 and dimP s. Inparticular, LGOs(V) is satisfied if the Neron-Tate height of an optimal singletoninV with defect at mosts is bounded from above uniformly.

Proof. Observe thatX(dimP) = 0 if and only ifP= X. In this case{P}canonly be an optimal singleton ofV if(V) = dim X. This equality entails V ={P}in which case the claim is trivial.

If X(dimP) > 0 the claim is direct consequence of Proposition 9.1 and the

definition ofLGOs(V).

The main result of this section states that a lower bound for the Galois orbit as in(8.1) is sufficient to prove that there are only finitely many optimal subvarieties ofV. Although we believe (8.1) to always hold, we are not able to prove it. However,we can show unconditionally that Opt(V; S) is finite for any fixed S 0.

Theorem 9.8. LetXbe an abelian variety defined over a fieldKwhich is finitelygenerated overQ. LetV Xbe a subvariety defined overK.

(i) Sayr, s 0 and suppose that all quotients ofXdefined over a finite exten-sion ofKsatisfyLGOrs . Then

(9.3) {A Opt(V); codimVA r and dimA dimAgeo s}

is finite.(ii) IfKis a number field, thenOpt(V; S) is finite for allS 0.

We obtain 2 immediate corollaries.

Corollary 9.9. Let us suppose that the height bound in Proposition 9.7 holds forall subvarieties of all abelian varieties defined over any number field. Then theZilber-Pink Conjecture2.2holds for all subvarieties of all abelian varieties definedover any number field.


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Corollary 9.10. LetXandVbe as in Theorem9.8. We suppose that all quotientsofXdefined over a finite extension ofKsatisfyLGOrs for allr, s 0. ThenOpt(V)is finite for any subvarietyV ofXdefined overK.

Let us look more closely at the case when Kis a number field andsis small. TheNeron-Tate height of a torsion point vanishes. So by Proposition 9.7any abelianvariety over a number field satisfies LGOr0 for all r 0. Part (i) of the theoremimplies thatVcontains only finitely many maximal torsion cosets as such subva-rieties are necessarily optimal. We recover the conclusion of the Manin-MumfordConjecture. Of course in this special case, our argument does not differ significantlyfrom the Pila-Zannier approach[35]. But part (i) of our theorem applied to s = 0even yields the following corollary.

Corollary 9.11. Let X and V be as in Theorem 9.8(i) with K a number field.Then

{A Opt(X); A= Ageo}

is finite.

The next case is s = 1; the corresponding case of the Zilber-Pink Conjectureconcerns subvarieties of codimension 2. So say {P} V is an optimal singletonwith dimP 1. If dimP= 0, then P is of finite order and we are back in thecase s = 0. So we assume dimP = 1. We know that {P} is geodesic-optimalwith respect to V by Proposition 4.5. In other words, P is not contained in acoset of positive dimension contained completely in V. In this setting it wouldbe interesting to know if h(P) is bounded from above in terms of V only. Theanalogous statement in the context of algebraic tori was proved by Bombieri andZannier, cf. Theorem 1[50]. Moreover, Checcoli, Veneziano, and Viada[14] showeda related result inside a product of elliptic curves with complex multiplication.

Proof of Theorem9.8. We can almost prove parts (i) and (ii) simultaneously. How-ever, at times we will branch off the main argument to specialise to the two state-ments. The proof will be by induction on dim V. Our theorem is trivial ifV is apoint. Say dim V 1. After replacing K by a finite extension we may supposethat V, all abelian subvarieties ofX, and all relevant homomorphisms below aredefined overK. We may assume thatKis a subfield ofC. We fix L an ample linebundle onXto make sense of the complexity ().

Suppose A is an element of Opt(V) or Opt(V; S) depending on whether we arein case (i) or (ii) of the theorem.

In case (i) we suppose codimVA r and dimA dimAgeo s; in case (ii)we suppose that A contains a point of height at most (2[K(A) : K])S.

By Proposition4.5 the subvariety A is geodesic-optimal and thus an irreduciblecomponent ofV Ageo. By Proposition6.1 the coset Ageo is the translate of

an abelian subvariety Y X that comes from a finite set depending only on V.We observe that this finiteness statement is trivial if dim V= 1; we do not requireProposition6.1ifVis a curve. We will also fix an ample and symmetric line bundleon the abelian variety X/ Y in order to speak of the Neron-Tate heighth.

Let : X X/Y be the canonical morphism. As we are in characteristic 0,there is a Zariski open and dense subset V V such that |V :V (V) is a

smooth morphism of relative dimension n and (V) is Zariski open in (V), cf.Corollary III.10.7[22].


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IfA V = , then A is contained in an irreducible component ofV V andA is an optimal subvariety of this irreducible component. In both case (i) and (ii)we may apply induction on the dimension as dim(V V) < dim V; for case (i)we observe that the codimension in (9.3) drops. So there are only finitely manypossibilities for A.

Let us assumeA V =. We note that A V is an irreducible component of afibre of|V . The fibres of|V are equidimensional of dimension n. So dim A= nand maps A to someP (X/Y)(C).

SincePlies in the torsion coset (A) we find P (A) and thus1(P)1((A)) A + Y . But Y is contained in a translate of A and thusA + Y =A. We conclude

(9.4) dim Y+ dimP= dim 1(P) dimA.

Next we claim that the singleton {P} is an optimal subvariety of (V). If the contrary holds there is a subvariety B of (V) containing P, with positive

dimension, and defect at most dimP. We fix an irreducible component B, thatmeets V, of the pre-image ofB under |V with A B. As |V is smooth ofrelative dimension n = dim A we have

(9.5) dim B= dim B + dim A > dim A.

We remarkB 1(B), so dimB dim Y+dimB. Since(B) dimPwe find

dimB dim Y+ dim B + dimP.

Optimality ofA and B A from (9.5) imply

dimA< dim A + dimB dim B dim A + dim Y+ dim B dim B+ dimP.

We use the equality in (9.5) to find dimA < dim Y+ dimP. This contradicts(9.4) and so{P}must be an optimal subvariety of(V).

Let us suppose dim A >0 for the moment. Then dim (V) = dim V dim A


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34 P. HABEGGER AND J.

o-minimality and certain atypical intersections

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