Rational points in periodic analytic sets and the Manin-Mumford conjecture
aa r X i v : . [ m a t h . N T ] F e b Rational points in periodic analytic sets and the Manin-Mumford conjecture
Jonathan Pila & Umberto Zannier
Abstract.
We present a new proof of the Manin-Mumford conjecture about torsion points onalgebraic subvarieties of abelian varieties. Our principle, which admits other applications, is toview torsion points as rational points on a complex torus and then compare (i) upper bounds forthe number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii)lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to beable to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analyticvariety supposed invariant for translations by a full lattice, which is a topic with some independentmotivation. §
1. Introduction
The so-called
Manin-Mumford conjecture was raised independently by Manin and Mumfordand first proved by Raynaud [R] in 1983; its original form stated that a curve C (over C ) of genus ≥ , embedded in its Jacobian J , can contain only finitely many torsion points (relative of courseto the Jacobian group-structure). Raynaud actually considered the more general case when C isembedded in any abelian variety. Soon afterwards, Raynaud [R2] produced a further significantgeneralization, replacing C and J respectively by a subvariety X in an abelian variety A ; in thissituation he proved that if X contains a Zariski-dense set of torsion points, then X is a translateof an abelian subvariety of A by a torsion point . Other proofs (sometimes only for the case ofcurves) later appeared, due to Serre, to Coleman, to Hindry, to Buium, to Hrushovski (see [Py]),to Pink & Roessler [PR], to M. Baker & Ribet [BR]. We also remark that a less deep precedent ofthis problem was an analogous question for multiplicative algebraic groups, raised by Lang alreadyin the ’60s. (See [L]; Lang started the matter by asking to describe the plane curves f ( x, y ) = 0with infinitely many points ( ζ, η ), with ζ, η roots of unity.)In the meantime, the statement was put into a broader perspective, by viewing it as a specialcase of the general Mordell-Lang conjecture and also, from another viewpoint, of the Bogomolovconjecture on points of small canonical height on (semi)abelian varieties (we recall that torsionpoints are those of zero height). These conjectures have later been proved and unified (by Faltings,Vojta, Ullmo, Szpiro, Zhang, Poonen, David, Philippon,...) by means of different approachesproviding, as a byproduct, several further proofs of the Manin-Mumford statement (we refer to thesurvey papers [Py] and [T] for a history of the topic and for references.) Recent work of Klingler,Ullmo and Yafaev proving (under GRH) the Andr´e-Oort conjecture, an analogue of the Manin-Mumford conjecture for Shimura varieties, has inspired another proof of Manin-Mumford due toRatazzi & Ullmo [RU]. All of these approaches are rather sophisticated and depend on tools ofvarious nature.It is the purpose of this paper to present a completely different proof compared to the ex-isting ones. Our approach too relies on certain auxiliary results, having however another nature(archimedean) with respect to the prerequisites of the previously known proofs; hence we believethat this treatment may be of some interest for a number of mathematicians. Also, and perhapsmore importantly, the underlying principle has certainly other applications, as in work in progress[MZ]; we shall say a little more on this at the end.1n short, the basic strategy of our proof is as follows: view the torsion points as rational pointson a real torus; estimate from above the number of rational points on a transcendental subvariety(Pila-Wilkie); estimate from below the number of torsion points by considering degree and takingconjugates (Masser); obtain a contradiction if the order of torsion is large. In more detail, wecan proceed along the following steps, sticking for simplicity to the case of an algebraic curve X ,of genus ≥
2, in the abelian variety A , both over a number field. (The general case of complexnumbers can be dealt with in a similar way or reduced to this by specialization.)(i) There is a complex analytic group-isomorphism β : C g / Λ → A , where Λ is a certain latticeof rank 2 g , say with a Z -basis λ , . . . , λ g ; so we can view a torsion point P ∈ A as the image P = β ( x ) where x is the class modulo Λ of a vector r λ + . . . + r g λ g ∈ C g , where the r i arerationals; if P has exact order T , the r i will have exact common denominator T .(ii) The algebraic curve X ⊂ A equals β ( Y ), where Y = β − ( X ) is a complex analytic curvein C g / Λ; in turn, if π : C g → C g / Λ is the natural projection, we may write Y = π ( Z ), where Z = π − ( Y ) ⊂ C g is an analytic curve which is invariant under translations in Λ.(iii) We can use the basis λ , . . . , λ g to view C g as R g ; then Z will become a real analyticsurface in R g , denoted again Z , invariant under Z g . Also, as in (i), the torsion points on A willcorrespond to rational points in R g . Then, the torsion points on X will correspond to rationalpoints on Z . Note that, in view of the invariance of Z by integral translations, it suffices to studythe rational points on Z in a bounded region of R g .(iv) Due to a method introduced by Bombieri-Pila [BP] for curves, and further developed byPila [P] for surfaces, and by Pila-Wilkie [PW] in higher dimensions, one can get good estimates forthe number of rational points with denominator T on a bounded region of a real analytic variety.As has to be expected, these estimates apply only if one confines the attention to the rationalpoints which do not lie on any of the real semi-algebraic curves on the variety. For the numberof these points, the estimates take the shape ≪ T ǫ , for any given ǫ >
0. This can be applied tothe above defined Z ; it turns out that, since X is not a translate of an elliptic curve, Z does notcontain any semi-algebraic curve.(v) All of this still does not still yield any finiteness result, merely estimates for the number oftorsion points on X , of a given order T . The crucial issue is that if X contains an algebraic point P , it automatically contains its conjugates over a field of definition. Now, Masser has proved in1984 that the degree, over a field of definition for A , of any torsion point of exact order T is ≫ T ρ for a certain ρ > A . Hence, if X contains a point of order T , it containsat least ≫ T ρ such points.Then, comparing the estimates coming from (iv) and (v) we deduce that the order of thetorsion points on X is bounded, concluding the argument.In some previous proofs, one exploited not a lower bound for degrees, but rather the Galoisstructure of the field generated by torsion points. However this may be considered an informationsubstantially of different nature.To better isolate the new arguments from previous ones, we will prove the Manin-Mumfordconjecture in the following weak form, which includes the case of curves. Theorem 1.1.
Let A be an abelian variety and X an algebraic subvariety of A , both defined overa numberfield. Suppose that X does not contain any translate of an abelian subvariety of A ofdimension > . Then X contains only finitely many torsion points of A . Theorem 1.1*.
Let A be an abelian variety and X an algebraic subvariety of A , both definedover a numberfield. The Zariski closure of the set of torsion points of A contained in X is a finiteunion of torsion cosets. Concerning the proof-method, a further point is that, in order for the Pila-Wilkie estimatesto be applicable, we must study the real semi-algebraic curves that may lie on the inverse image Z of X under the analytic uniformization C g → A . The set Z is analytic (defined by the vanishingof certain polynomials in the abelian functions giving the map C g → A ) and periodic modulo thelattice Λ.In §
2, Theorem 2.1, we show that a connected real semi-algebraic curve contained in such aset Z must be contained in a complex linear subspace contained in Z , and moreover a subspacein which the period lattice has full rank. In other words, we prove that the “algebraic part” of Z in the sense of Pila-Wilkie (see Def. 2.1 below) corresponds precisely to the union of translates ofabelian subvarieties of A of dimension > X . This fact seems to be not entirelyfree of independent interest. Remark.
One can also check that our proof of Theorem 2.1 works in fact for arbitrary complextori C g / Λ, even if they are not projective.With this result in hand the proof of Theorem 1.1 is concluded in § §
2. Structure of the algebraic part of a periodic analytic set
As announced, this section will be devoted to a complete description of algebraic subsets of ananalytic subvariety of a complex torus, which corresponds to an analytic subvariety of C g periodicfor a full lattice Λ, that is, invariant by translations in Λ. We express explicitly all of this in a fewdefinitions and then state the main Theorem 2.1.We take a (full rank) lattice Λ ⊂ C g which will be fixed throughout. We further fix a Z -basis λ , . . . , λ g of Λ and use it to identify C g with R g .We shall use throughout the notation Z + Λ := ∪ λ ∈ Λ ( Z + λ ) := { z + λ : z ∈ Z, λ ∈ Λ } forthe union of the translates of Z by the vectors in Λ. A set Z ⊂ C g will be called ( Λ -) periodic if Z + Λ = Z .As usual, an analytic set Z ⊂ C g will mean a set such that every point z ∈ C g has an openneighbourhood U in which Z is defined as the set of common zeros of a finite collection of functions(depending on z ) that are (complex) analytic (i.e. regular) in U . Such a set is readily seen to be real analytic as a subset of R g . Definition 2.1.
Let Z ⊂ C g . We define the complex algebraic part Z ca of Z to be the union ofconnected closed algebraic subsets of C g of positive dimension contained in Z . Viewing Z as asubset of R g we define the real algebraic part Z ra of Z to be the union of connected real algebraicsets of positive dimension contained in Z . Finally we define the algebraic part Z alg of Z to bethe union of all connected real semi-algebraic sets of positive dimension (see [Sh, pp. 51, 100] fordefinitions) contained in Z . One readily sees that Z ca ⊂ Z ra ⊂ Z alg .3e reserve the term subspace for a (respectively complex or real) vector subspace of C g or R g .By a (respectively complex or real) linear subvariety we mean a subset of C g or R g defined by thevanishing of some (complex or real) linear equations, not necessarily homogeneous. A subspace H in which H ∩ Λ has full rank in H will be called a ( Λ -) full subspace. A set of the form z + H where z ∈ C g and H is a complex linear subspace will be called a coset . Thus a subspace H thatis both complex and full corresponds precisely to a subtorus of C g / Λ, and a coset of such an H will be called a torus coset . Definition 2.2.
For a set Z ⊂ C g we let Z torus coset be the union of torus cosets of positivedimension contained in Z .A torus coset is evidently a complex linear subvariety, whence Z torus coset ⊂ Z ca ⊂ Z ra ⊂ Z alg .For a periodic analytic set Z we show that all these sets coincide. Theorem 2.1.
Let Z ⊂ C g be a periodic analytic set. Then Z alg = Z torus coset . Our proof of this result involves several preliminaries; for the reader’s convenience we explicitlysubdivide the proof into three Steps.
Step 1.
Reduction semi-algebraic → complex algebraic. In this step we reduce to complexcurves contained in Z , by proving that Z alg = Z ca . We first prove a lemma: Lemma 2.1.
Let Z ⊂ C g be analytic. Suppose that x ∈ C g has a neighbourhood U such that x is a smooth point of Y ∩ U , where Y is a real algebraic curve with Y ∩ U ⊂ Z as subsets of R g .Then there is a neighbourhood U ′ of x contained in U and a finite collection of irreducible complexalgebraic curves Γ j such that Y ∩ U ′ ⊂ ∪ j Γ j ⊂ Z .Proof. We took coordinates in R g using the lattice basis, but clearly Y will remain semi-algebraicunder any real linear change of coordinates. Also, for our purposes we can assume that Y isirreducible.Let us here write ( x , y , . . . , x n , y n ) for the coordinates of R g , where z j = x j + iy j are thecoordinates of C g . We may assume by translation that x = 0. If x (say) is a non-constant functionon the real curve Y near 0 then, for t = x in some real neighbourhood of 0, all the functions x , y , . . . , x n , y n are real analytic functions of some m -th root u := t /m , and (as functions of u ), are algebraic over R ( x ) ⊂ C ( x , y ). We claim that each of the functions x , y , . . . , x n , y n isalgebraic over the field C ( z ) = C ( x + iy ). This is because the function x + iy is non-constanton Y since x is non-constant, and C ( x , y ) is algebraic over C ( x ), so of transcendence degree 1.Therefore, for each j , the function z j = x j + iy j is also algebraic over C ( z ). The functions z j ( u ) areanalytic for real u in some neighbourhood of 0, hence for complex u in a complex neighbourhoodof 0; the image of u z ( u ) is thus a complex open subset of a complex algebraic irreducible curveΓ, which in a neighborhood of 0 contains Y and must necessarily be contained in Z . CertainlyΓ contains a smooth point in this neighborhood, and we get Γ ⊂ Z by analytic continuation, asthe smooth points of an irreducible complex curve are connected, and the remaining points areisolated and belong to Z by continuity.We can now prove the alluded reduction: Proposition 2.1.
Let Z ⊂ C g be a periodic analytic set. Then Z alg = Z ca .Proof. Suppose W is a connected real semi-algebraic set of positive dimension with W ⊂ Z . Then,omitting at most finitely many points, W is a union of real connected semi-algebraic curves. If Y is4uch a curve then, Y is real algebraic in the neighbourhood of any smooth point and we may applyLemma 2.1 to find Y contained in a complex algebraic curve Γ ⊂ Z . This proves the statement. Step 2.
Complex curves in periodic analytic varieties and linear spaces.
In this step weprove that if a complex curve C is contained in the periodic analytic set Z , then there are ‘several’complex lines l with C + l contained in Z . In turn, this will involve a few preliminaries.Suppose C ⊂ C g is an irreducible complex algebraic curve. Of the coordinates z , . . . , z g of C g , if z i is not constant on C then any z i , z j are related by some irreducible polynomialequation G ( z i , z j ) = 0. For | z i | sufficiently large, say | z i | > R , the solutions z j are givenby convergent Puiseux series φ j ( z i ). By a branch of C we mean a choice of index i and a g -tuple φ = ( φ ( z i ) , . . . , φ g ( z i )) of algebraic Puiseux series, convergent for | z i | > R , and such that φ ( z i ) = ( φ ( z i ) , . . . , φ n ( z i )) ∈ C for all | z i | > R . By a suitable choice of index i , we can alwaysobtain a branch of C such that, for all j , φ j ( z i ) = α j z i + lower order terms , α j ∈ C . We call such a branch linear , and α = ( α , . . . , α g ) the direction of the branch. Observe that α i = 1, so α = 0.Suppose φ ( w ) = ( φ ( w ) , . . . , φ g ( w )) is an n -tuple of algebraic Puiseux series, convergent for | w | > R . Then, fixing w , w ∈ C and µ = ( µ , . . . , µ g ) ∈ C g , we consider (for κ ∈ C such that | w + κw | > R ), κ ψ ( κ ) = ( ψ ( κ ) , . . . , ψ g ( κ )) ,ψ i ( κ ) = φ i ( w + κw ) − κµ i . From the algebraic relation G i ( t, φ i ( t )), we find that G i ( w + κw , ψ i ( κ )+ κµ i ) = 0. Thus the ψ i ( κ )are also algebraic functions of κ , and the locus above is Zariski dense in an irreducible algebraiccurve in C g , which we denote Γ( φ, µ, w , w ). It is important to note that its degree is bounded interms of the degree of the curve containing φ (independently of the choice of w , w , µ ). We nowhave a lemma, perhaps known but for which we have found no reference: Lemma 2.2.
Let Z ⊂ C g be an analytic set, B ⊂ C g a bounded set, and δ a positive integer.There exists a positive integer K = K ( Z, B, δ ) with the following property. Suppose Γ ⊂ C g is anirreducible complex algebraic curve of degree ≤ δ , and with Z ∩ B ∩ Γ) ≥ K . Then Γ ⊂ Z .Proof. About each point of C g there is an open disk in which Z is defined by the vanishing of afinite number of regular functions. Taking a smaller disk, these functions may be assumed regularin a neighbourhood of the closure of the disk. Then by compactness of the closure of B , we canfind a finite number of open disks U covering B , and on each disk a finite number of functions,regular in a neighbourhood of the closure of the disk, whose zero-locus defines Z in the disk. Thus,in each of the finitely many disks, Z is expressed as in intersection of finitely many hypersurfaces.As shown in a moment, this remark reduces the proof to the case of hypersurfaces, namely to thefollowing Claim.
Suppose that U ⊂ C n is a bounded open disk and that f is a complex-valued function thatis regular in a neighbourhood of the closure of U . Let Y = { z ∈ U : f ( z ) = 0 } , and δ a positiveinteger. There is a positive integer K = K ( U, f, δ ) with the following property. Suppose Γ ⊂ C n isan irreducible curve of degree ≤ δ and Y ∩ C ≥ K . Then f vanishes identically on Γ in U . U in the finite coveringtake K U to be the maximum of the K ( U, f, δ ) over all the finitely many functions f defining Z on U , and take K = P U K U + 1. For each U put Z U = Z ∩ U . Let now Γ be an irreducible complexalgebraic curve of degree ≤ δ with Z ∩ B ∩ Γ ≥ K . By the pigeonhole principle, one of the disks U has Z U ∩ Γ) ≥ K U . By the Claim, each of the functions f defining Z on U vanishes identicallyon Γ, so that, restricted to U , Γ ⊂ Z . However Γ is connected, so by analytic continuation weconclude that Γ ⊂ Z . It remains then to prove the Claim, as we shall now do.Since Γ has degree ≤ δ , each pair of coordinates z i , z j satisfy on Γ an algebraic relation P ij ( z i , z j ) = 0 for some nonzero polynomial P ij ∈ C [ x , x ] of degree ≤ δ . These polynomials,considered up to nonzero constant factors, define an algebraic set of dimension 1, containing Γ asa component. This set is possibly reducible but, given the P ij , by projecting to a general plane, wesee that there certainly exists a hypersurface H of degree ≤ gδ that contains Γ but not any othercomponent of dimension 1; this H corresponds to a further polynomial P H of degree ≤ gδ . Now,the polynomials in two variables of degree ≤ δ , up to constants, are parametrised by a projectivespace P D , D = ( δ + 1)( δ + 2) /
2, whereas H is parametrized by P D ′ for a D ′ depending only on g, δ . Hence the set { H, P ij } is parametrized by the product∆ = P ( C ) D ′ Y ij P ( C ) D , a compact space, which we can assume contained in R m for some suitable m . For w ∈ ∆ let I w bethe ideal generated by the corresponding P ij , P H . Consider the set V := { ( z, w ) ∈ C g × ∆ : z = ( z , . . . , z g ) ∈ U, w ∈ ∆ , f ( z ) = 0 , Q ( z ) = 0 , all Q ∈ I w } . We have V ⊂ C g × ∆ with projections π , π onto the factors C g and ∆ respectively.Our very construction shows that every algebraic curve Γ ⊂ C g of degree ≤ δ is defined, up tofinitely many points, by I w for (at least one) suitable w = w Γ ∈ ∆; hence, for some point w Γ ∈ ∆, π (cid:0) π − ( w Γ ) (cid:1) equals the intersection Y ∩ Γ plus a finite set.Now we consider V as a subset of R M for suitable M . Since f is regular on a neighbourhoodof the closure of U , the set V is subanalytic (even semianalytic ) in R M (for the definition see [G]or [BM, Definition 2.1]). Further, V is bounded, since U is bounded and ∆ is compact. We appealto Gabrielov’s Theorem [G, or see e.g. BM, Theorem 3.14] to conclude: As w varies over the bounded set π ( V ) , the number of connected components of π − ( w ) isbounded by some finite number N = N ( U, f, D ) ; the number of connected components of π ( π − ( w )) is then also bounded by N . Put K = N + 1. If now Y ∩ Γ ≥ K , Y ∩ Γ cannot consist of isolated points and must, asa semianalytic set, have dimension ≥
1. This set must then contain some smooth real-analyticarc. If we take a point on such an arc that is a non-singular point of Γ, then restricting f to aneighbourhood in which we can complex analytically parameterize Γ, we find that f restricted tothis local parameterization is an analytic function with non-isolated zeros. It therefore vanishesidentically on Γ in this neighbourhood. But now since Γ is irreducible, the set of its non-singularpoints is connected, and we find that f vanishes identically on Γ in U by analytic continuation.This establishes the Claim, and concludes the proof of Lemma 2.2.6he following result now follows easily: Corollary.
Let Z ⊂ C g be a periodic analytic set and r > . Let φ ( w ) be an n -tuple of convergent(for | w | > R ) algebraic Puiseux series. There is an integer K = K ( Z, r, φ ) with the followingproperty. Let B ⊂ C g be a ball of radius r , and w , w ∈ C , τ, µ ∈ C g . If (cid:0) Γ( φ, µ, w , w ) + τ ∩ Z ∩ B (cid:1) ≥ K then Γ( φ, µ, w , w ) + τ ⊂ Z .Proof. For a fixed B the conclusion follows directly from Lemma 2.2, because Γ( φ, µ, w , w ) + τ is an algebraic curve of degree bounded only in terms of φ . Since Z is periodic, the ball B may beassumed to be centred in a fundamental domain, so dependence on the centre of the ball may beeliminated.We now come to the fundamental point of this Step 2. In the following, || · || denotes theEuclidean norm in C g . For a point z = ( z , . . . , z g ) ∈ C g − { } we denote by [ z ] the complex line { wz : w ∈ C } . Proposition 2.2.
Let Z ⊂ C g be a periodic analytic set and C be an irreducible complex algebraiccurve with C + τ ⊂ Z for a τ ∈ C g . Let φ ( w ) be a linear branch of C (convergent for | w | > R )with direction [ α ] and put K = K ( Z, , φ ) , as in the last Corollary. Finally, let λ ∈ Λ be such that || β − λ || < (2 K ) − for some β ∈ [ α ] . Then C + τ + [ λ ] ⊂ Z .Proof. Fix w ∈ C such that || w α − λ || < (2 K ) − . Now choose R ≥ R such that, for any w ∈ C with | w | > R + K | w | and any w ′ ∈ C with | w ′ | ≤ K | w | we have || φ ( w + w ′ ) − φ ( w ) − w ′ α || < / . This is possible by the condition on φ that all terms apart perhaps from the leading term aresub-linear. If now | w | > R + K | w | and k = 0 , , . . . , K we have || φ ( w + kw ) − φ ( w ) − kλ || ≤ || φ ( w + kw ) − φ ( w ) − kw α || + k || w α − λ || < . Now, φ ( w + kw )+ τ − kλ + ∈ Z because Z is periodic and C + τ ⊂ Z ; also, φ ( w + kw )+ τ − kλ ∈ Γ := Γ( φ, λ, w , w ) + τ provided | w + kw | > R , and by the above all these points lie in the ballof radius 1 about φ ( w ) + τ for k = 0 , , . . . , K .By the Corollary to Lemma 2.2, we deduce that Γ ⊂ Z .We thus find Γ( φ, λ, w , w ) + τ ⊂ Z for all w suitably large, but we must still show that infact all C + τ + [ λ ] ⊂ Z . If we set y = w + κw , κ ∈ C , we find that φ ( y ) + τ − (( y − w ) /w ) λ ∈ Z provided | y | > R . But, fixing y, κ, w , the above represents a line as w varies. Since a segment ofthis line lies in Z , the whole line does, by analytic continuation. So for x ∈ C the points φ ( y ) + τ − xλ ∈ Z provided only | y | > R . Fixing now x , the curve in y is just a branch of C + τ − xλ . This curve isirreducible, being a translate of C , and so C + τ − xλ ⊂ Z for all x .7 tep 3. Linear subvarieties of a periodic analytic variety.
We study general linear subvarietiesof a periodic analytic variety Z , their intersections with the lattice Λ, and use all of this to exploitthe important conclusion of the last proposition, which produces certain translates C + l of C bya line, contained in Z .We first recall some useful simple facts from the known theory of closed subgroups of realvector spaces, and start by noting that: if H is a real subspace of C g then the closure of H + Λ has the form K + Λ where K is a full real subspace containing H . This statement follows from the description of closed subgroups of real vector spaces givenin [S, Lecture VI, § C g of H + Λ has the form K + Λ for K a real subspace and Λ a lattice in a space W complementary to K . Clearly H ⊂ K .Now, the projection of Λ to W (along K ) is Λ , which must then have full rank in W , because R g = R Λ ⊂ R ( K + Λ ) = K + R Λ . Now, lifting a basis of Λ to Λ we see that Λ := Λ ∩ K hasmaximal rank, i.e. equal to dim K , as wanted.We further note that for any open ball I around in K , the set H + I contains a set ofgenerators for Λ := Λ ∩ K . To prove this, let Λ ′ denote the lattice generated by Λ ∩ ( H + I ) andobserve that H + Λ ′ is dense in K : in fact, by definition of Λ ′ , the closure of H + Λ ′ contains theintersection of I/ H + Λ, so it contains I/
2, whence it must contain the whole K . Now, let λ ∈ Λ ; it belongs to the closure K of H + Λ ′ ; hence λ − I intersects H + Λ ′ , so λ + Λ ′ intersects Λ ∩ ( H + I ) ⊂ Λ ′ , proving that λ ∈ Λ ′ and so Λ ⊂ Λ ′ , as desired.Now we may give the following definitions: Definition 2.3.
Let H be a real subspace of C g .1. We denote by c( H ) := C H the complex space generated by H and call it the complexclosure of H ; we call H complex if H = c( H ).2. We denote by f( H ) the full closure of H , namely the full real subspace K such that H + Λis dense in K + Λ, as in the above remark. So H is full just if f( H ) = H .3. We denote by fc ( H ) the full-complex closure of Z , namely the union of the iterates of H under the map H f(c( H )).One could alternately iterate the two operations H c( H ), H f( H ). In general, if g > H )) = c(f( H )). (In C take for instance H = R (1 , Z (1 ,
0) + Z ( i,
1) + Z (1 , i ) + Z (1 , √ f ( H ) = H , c ( H ) = C (1 , c ( H ) is not full, we have that f c ( H ) contains properly cf ( H ) = c ( H ).) Anyway fc ( H ) may also be characterized as the smallestsubspace containing H that is both full and complex. Lemma 2.3.
Let Z ⊂ C g be a periodic analytic set. Suppose z ∈ C g and H is a real subspace of C g with z + H ⊂ Z . Then there is a torus coset z + M with z + H ⊂ z + M ⊂ Z , and one cantake M = fc ( H ) .Proof. Note that by Def. 2.3(2) (which amounts to the opening remarks of this Step 3), if H isa real subspace and z ∈ C g , then ( z + H ) + Λ is dense in some ( z + K ) + Λ where K is a fullreal subspace containing H . Now, if H is a real subspace with z + H ⊂ Z then, by periodicity,the fact just stated and continuity, we have z + f( H ) ⊂ Z , and, by analytic continuation, we have z + c( H ) ⊂ Z . The conclusion follows by applying these observations to H and its iterates underfull and complex closure.We can now rapidly conclude the proof of Theorem 2.1; we prove a last lemma:8 emma 2.4. Let Z ⊂ C g be a periodic analytic set, C an irreducible complex algebraic curve and M a complex subspace such that C + M ⊂ Z . Suppose that C + M is not a coset of M . Then thereis a complex subspace M ′ with M ⊂ M ′ , dim M < dim M ′ , and C + M ′ ⊂ Z .Proof. Let N be a complex linear subspace supplementary to M in C g . So every translate z + M ,where z ∈ C g , intersects N in just one point. Now, C + M , as the image of the sum-map C × M → C + M contains an open dense set in its Zariski closure in C g , which is an irreduciblealgebraic subvariety of C g ; also, ( C + M ) ∩ N is irreducible (because it is the projection of C + M to N along M ). By hypothesis ( C + M ) ∩ N is not equal to a point, hence it cannot be a finiteset of points and therefore it contains an open dense subset of a closed irreducible algebraic curve C ′ ⊂ N ; note that C + M ⊂ C ′ + M ⊂ Z and dim( C + M ) = dim M + 1.Take now a linear branch φ of C ′ with direction [ α ], so [ α ] lies in N . Then there is a smallopen ball B around 0 in C g so that ([ α ] + B ) ∩ M does not contain any nonzero lattice point. Bya remark above, Λ ∩ ([ α ] + B ) contains generators for the full closure of [ α ], so in particular itcontains a lattice point λ M .Put M ′ = M + [ λ ].Then C ′ + M ′ ⊂ Z . For if τ ∈ M we have C ′ + τ ⊂ Z , but this curve is a translate of C ′ , andby Proposition 2.2 (applied to C ′ in place of C ) we find that for small enough B , ( C ′ + τ )+ [ λ ] ⊂ Z .This being true for all τ ∈ M , we have C ′ + M ′ ⊂ Z . Now so C + M ′ ⊂ C ′ + M ′ ⊂ Z . Thiscompletes the proof. Proof of Theorem 2.1
Let C be an irreducible complex algebraic curve contained in Z and take a maximal complexsubspace M such that C + M ⊂ Z . By maximality and Lemma 2.4 we deduce that C + M is asingle coset of M , which must be a torus coset by maximality and Lemma 2.3.It follows that every closed complex algebraic set of positive dimension contained in Z is con-tained in the union of full cosets of positive dimension contained in Z , i.e. that Z ca ⊂ Z torus coset .Proposition 2.1 now finally proves what we need. Remark.
Note that the final argument easily leads to the following assertion:
If an irreduciblecomplex algebraic set V is contained in Z then there is a torus coset z + M with V ⊂ z + M ⊂ Z . To prove this, take a maximal torus M such that for a point z we have z + M ⊂ Z . Applying thelast conclusion of the proof of Theorem 2.1 to all the irreducible curves C on V passing through z we see by maximality that z + M contains a neighborhood of z in V . Hence it contains V . §
3. Manin-Mumford
Let now A be a (projective) Abelian variety, and X a subvariety of A . We have a complexanalytic uniformization C g → A , periodic with period lattice Λ. The preimage Z ⊂ C g of X is aperiodic analytic set, and we have shown that all real semialgebraic subsets, connected of positivedimension, of the real reduct of Z are contained in the union of torus cosets contained in Z .Now a subtorus of C g / Λ corresponds to an abelian subvariety of A (see e.g. [R, remark onpage 86], or it may be argued directly using Chow’s Theorem that such a subtorus is algebraic). Proof of Theorem 1.1.
Our interest is in the torsion points P of A that lie on X . Let tor( A ) denotethe torsion subgroup of A , consisting of all points of A of finite order. The order T = T ( P ) of P isthe minimal positive integer with T P = 0. A torsion point P of A corresponds to a rational point z = z P = ( q , . . . , q g ) ∈ Q g of Z considered as a subset of R g . The order of P is equal to the9 enominator of z , i.e. the minimal integer d > dz ∈ Z g . By the present assumptions, X does not contain any translate of an abelian subvariety of dimension >
0; hence, by Theorem2.1 the set Z alg = Z torus coset is empty.As a subset of R g the set Z is Z g -periodic. In considering torsion points it therefore sufficesto replace Z by Z = Z ∩ [0 , g , and clearly Z alg = Z alg ∩ [0 , g is empty as well.For a set W ⊂ [0 , g and a real number T ≥ N ( W, T ) the number of rationalpoints of W of denominator dividing T . By Pila-Wilkie [PW], for every ǫ > N ( Z , T ) = N ( Z − Z alg , T ) ≤ c ( Z , ǫ ) T ǫ . (3 . A is definedover a number field K . For P ∈ tor( A ) set d ( P ) = [ K ( P ) : Q ]. Then Masser [M] proves that d ( P ) ≥ c ( A ) T ρ for some c ( A ) > ρ > g of A . Note that allthe conjugates of P over a number field of definition for both A, X are still torsion points on X , ofthe same order as P . By the lower bound of Masser just displayed, the number of such conjugatesis at least c ( A ) T ρ . Hence we get at least c ( A ) T ρ distinct points z ∈ Z of height ≤ T , corresponding to P and its conjugates. Choosing in (3.1) ǫ = ρ/ T is bounded.As anticipated in the Introduction, we note that the same arguments may be used to provethe stronger version given by Theorem 1.1*, at the cost of adding some simple geometrical factson subvarieties of abelian varieties and cosets contained in them. Let us briefly recall this here.The point is to describe the cosets contained in X . Call such a coset b + B ( b a point, B anabelian subvariety of A ) maximal if it is not included in any other larger torus coset still containedin X . Then one may prove that the set of abelian subvarieties B is finite, for b + B runningthrough maximal cosets. A simple proof of this is in [BZ], Lemma 2. (It uses only rather standardconsiderations involving degrees.) With this in mind, we proceed to illustrate the proof of Theorem1.1* by induction on dim A , the case of dimension 0 (or 1) being indeed trivial. Using exactly thesame method of proof of the weaker version, it suffices, through Theorem 2.1, to deal with thetorsion points which lie in some coset b + B contained in X and having dimension >
0. We maythen assume that this coset is maximal, and thus that B lies in a certain finite set. Thus for ourpurposes we may assume that B is fixed. The quotient A/B has the structure of an abelian variety A ′ and we have dim A ′ < dim A . Let π : A → A ′ be the natural projection. The set of b ∈ A suchthat b + B is contained in X is easily seen to be an algebraic variety, which projects to a variety X ′ under π . The coset b + B contains a torsion point if and only if π ( b ) is torsion on A ′ . Now, theinduction assumption applied to A ′ , X ′ easily concludes the proof. Final remarks.
1. The Manin-Mumford conjecture for A and X defined over C follows from the above byspecialization arguments, as in the original papers of Raynaud. Moreover, it is known that aversion that is uniform (regarding the number of cosets required to contain all the torsion points)10s X varies over a family of subvarieties of fixed dimension and degree of a given abelian varietyfollows from the above version, as described in [BZ] or see Hrushovski [H] or Scanlon [Sc] (so-called“automatic uniformity”).2. It seems that an argument along the present lines can also be given for the easier multi-plicative version of Manin-Mumford mentioned in §
1. Here the role of our Theorem 2.1 would beplayed by the theorem of Ax [Ax] establishing Schanuel’s conjecture for power series.3. Quantitative versions of the Manin-Mumford conjecture (and indeed of the Mordell-Langconjecture) giving upper estimates for the number of torus cosets are given by R´emond [Re]. Gooddependencies e.g. on the degree of X when A is fixed also follow from Hrushovski’s proof [H]. Ourmethod does not for the present yield any new quantitative information. The result of Pila-Wilkieholds in an arbitrary o-minimal structure. In this generality it seems one could not hope for gooddependence e.g. on the degree of X in the Pila-Wilkie result. For the specific case of sets definedby algebraic relations among theta-functions one might hope for good bounds, but we have notpursued this. For the multiplicative version, bounds with good dependencies on the degrees of thevarieties were obtained by R´emond [Re2]; see also Beukers-Smyth (for curves) and Aliev-Smyth(in general) [BS, AS].4. It seems clear that the constant c can be taken uniformly as X varies over all subvarietiesof A of fixed dimension and degree, by standard uniformity properties in o-minimal structures.5. In a paper in progress by Masser and Zannier, the present method has been applied toprove the following: For l = 0 , , let E l be the elliptic curve y = x ( x − x − l ) and let P l , Q l betwo points on E l with x -coordinate resp. 2,3. Then there are only finitely many complex values of l such that both P l , Q l are torsion on E l . This kind of result is related to Silverman’s specializationtheorem, a special case of which implies that the l ∈ C such that P l or Q l is torsion form a setof algebraic numbers of bounded height. The finiteness statement however seems not to followdirectly from any known result. (The Manin-Mumford statement, applied to a suitable subvarietyof E × E , would work if P l , Q l would be taken as variable Z -independent points on a fixed ellipticcurve E .) Acknowledgements.
It is a pleasure to thank David Masser for very helpful discussions and forgenerously providing us with the exact parts of his work most relevant here. We also thank MattBaker for his kind interest and indication of references. The first author is grateful to the ScuolaNormale Superiore di Pisa for support and hospitality during the preparation of the present paper,and the second author is similarly grateful to the University of Bristol.
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