A consequence of the relative Bogomolov conjecture
aa r X i v : . [ m a t h . N T ] S e p A CONSEQUENCE OF THE RELATIVE BOGOMOLOVCONJECTURE
VESSELIN DIMITROV, ZIYANG GAO, AND PHILIPP HABEGGER
Abstract.
We propose a formulation of the relative Bogomolov conjecture and showthat it gives an affirmative answer to a question of Mazur’s concerning the uniformityof the Mordell–Lang conjecture for curves. In particular we show that the relativeBogomolov conjecture implies the uniform Manin–Mumford conjecture for curves. Theproof is built up on our previous work [DGH20].
Contents
1. Introduction 12. Proof of the main result for F = Q
33. From Q to an arbitrary base field in characteristic 0 84. Relative Bogomolov for isotrivial abelian schemes 9References 111. Introduction
Let F be a field of characteristic 0. A smooth curve C defined over F is a geometricallyirreducible, smooth, projective curve defined over F . We denote by Jac( C ) the Jacobianof C . The following conjecture is a question posed by Mazur [Maz86, top of page 234]. Conjecture 1.1.
Let g ≥ be an integer. Then there exists a constant c ( g ) ≥ with thefollowing property. Let C be a smooth curve of genus g defined over F , let P ∈ C ( F ) ,and let Γ be a subgroup of Jac( C )( F ) of finite rank rk(Γ) . Then (1.1) C ( F ) − P ) ∩ Γ ≤ c ( g ) where C − P is viewed as a curve in Jac( C ) via the Abel–Jacobi map based at P . Two particular consequences of this conjecture are:(i) Take F a number field and Γ = Jac( C )( F ). By the Mordell–Weil TheoremJac( C )( F ) is a finitely generated abelian group. Then (1.1) becomes a bound onthe number of rational points C ( F ) ≤ c ( g ) C )( F ) .(ii) If F = C and Γ = Jac( C ) tor , then (1.1) becomes C ( C ) − P ) ∩ Jac( C ) tor ≤ c ( g ),the uniform Manin–Mumford conjecture for curves in their Jacobians.In a recent work, we proved (1.1) provided that the modular height of the curve inquestion is large in terms of g ; see [DGH20, Theorem 1.2]. Mathematics Subject Classification.
The goal of this note is to give a precise statement for the folklore relative Bogomolovconjecture , and prove that it implies the full Conjecture 1.1.1.1.
The Relative Bogomolov Conjecture.
We start by proposing a formulation forthe relative Bogomolov conjecture.Let S be a regular, irreducible, quasi-projective variety defined over Q . Let π : A → S be an abelian scheme of relative dimension g ≥
1. Let L be a symmetric relatively ampleline bundle on A /S .For each s ∈ S ( Q ), the line bundle L s on the abelian variety A s = π − ( s ) is symmetricand ample; note that A s is defined over Q . Tate’s Limit Process provides a fiberwiseN´eron–Tate height ˆ h A s , L s : A s ( Q ) → [0 , ∞ ); it vanishes precisely on the torsion pointsin A s ( Q ). Finally define ˆ h L : A ( Q ) → [0 , ∞ ) to be P ˆ h A π ( P ) , L π ( P ) ( P ).Let η be the generic point of S and fix an algebraic closure of the function field of S .For any subvariety X of A that dominates S , denote by X η the geometric generic fiberof X . In particular, A η is an abelian variety over an algebraically closed field. Conjecture 1.2 (Relative Bogomlov Conjecture) . Let X be an irreducible subvariety of A defined over Q that dominates S . Assume that X η is irreducible and not contained inany proper algebraic subgroup of A η . If codim A X > dim S , then there exists ǫ > suchthat X ( ǫ ; L ) := { x ∈ X ( Q ) : ˆ h L ( x ) ≤ ǫ } is not Zariski dense in X . The name
Relative Bogomolov Conjecture is reasonable: the same statement with ǫ = 0 is precisely the relative Manin–Mumford conjecture proposed by Pink [Pin05,Conjecture 6.2] and Zannier [Zan12], which is proved when dim X = 1 in a series ofpapers [MZ12, MZ14, MZ15, CMZ18, MZ18].The classical Bogomolov conjecture, proved by Ullmo [Ull98] and S. Zhang [Zha98a], isprecisely Conjecture 1.2 for dim S = 0. When dim S = 1 and X is the image of a section,Conjecture 1.2 is equivalent to S. Zhang’s conjecture in his 1998 ICM note [Zha98b, § A η is simple and is proved by DeMarco–Mavraki [DM20, Theorem 1.4] if A → S isisogenous to a fibered power of an elliptic surface. In general Conjecture 1.2 is still open.1.2. Main result.
Our main result, which is built up on [DGH20], is the followingtheorem.
Theorem 1.3.
The Relative Bogomolov Conjecture, i.e., Conjecture 1.2, implies Con-jecture 1.1.
The proof of Theorem 1.3 is as follows. First we reduce Conjecture 1.1 to the case F = Q by using a specialization result of Masser [Mas89]. This is executed in §
3. When F = Q , our proof follows closely and uses our previous work [DGH20], where we proved(1.1) for all curves whose modular height is bounded below by a constant dependingonly on g ; see [DGH20, Theorem 1.2]. The key point in [DGH20] is to prove a heightinequality, which we cite as Theorem 2.2 in the current paper. As is shown by theproof of [DGH20, Proposition 7.1], the extra hypothesis on the modular height of curvesrequired in [DGH20, Theorem 1.2] is necessary because of the constant term in thisheight inequality. In this paper, we show that this constant term can be removed if we CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 3 assume the Relative Bogomolov Conjecture; see Proposition 2.3 for a precise statement.Then by following the framework presented in [DGH20] we can prove our Theorem 1.3for F = Q . Acknowledgements.
Vesselin Dimitrov has received funding from the European Union’sSeventh Framework Programme (FP7/2007–2013) / ERC grant agreement n ◦ ◦ ◦ Proof of the main result for F = Q In this section we prove Theorem 1.3 when F = Q . Theorem 2.1.
The Relative Bogomolov Conjecture, i.e., Conjecture 1.2, implies Con-jecture 1.1 for F = Q . The proof follows closely and is based on our previous work [DGH20].2.1.
Basic setup.
Fix an integer g ≥
2. Let M g be the fine moduli space of smooth pro-jective curves of genus g with level-4-structure, cf. [ACG11, Chapter XVI, Theorem 2.11(or above Proposition 2.8)], [DM69, (5.14)], or [OS80, Theorem 1.8]. It is known that M g is an irreducible regular quasi-projective variety defined over Q , and dim M g = 3 g − C g over M g , it is smooth and proper over M g with fibers that are smooth curves of genus g . Moreover, it is equipped with level 4-structure.Let Jac( C g ) be the relative Jacobian of C g → M g . It is an abelian scheme equippedwith a natural principal polarization and with level-4-structure; see [MFK94, Proposi-tion 6.9].Let A g be the fine moduli space of principally polarized abelian varieties of dimension g with level-4-structure. It is known that A g is an irreducible regular quasi-projectivevariety defined over Q ; see [MFK94, Theorem 7.9 and below] or [OS80, Theorem 1.9].Here too we have a universal object, the universal abelian scheme π : A g → A g of fiberdimension g . There is a canonical relatively ample line bundle L on A g / A g satisfying[ − ∗ L = L ; see [Pin89, Proposition 10.8 and 10.9].Attaching the Jacobian to a smooth curve induces the Torelli morphism τ : M g → A g .The famous Torelli theorem states that, absent level structure, the Torelli morphism isinjective on C -points. In out setting, τ is finite-to-1 on C -points, cf. [OS80, Lemma 1.11].As A g is a fine moduli space we have the following Cartesian diagram(2.1) Jac( C g ) / / (cid:15) (cid:15) ❴✤ A gπ (cid:15) (cid:15) M g τ / / A g Suppose we have an immersion A g ⊆ P N Q defined over Q , such an immersion exists.We write A g for the Zariski closure of A g in P N Q . Then the absolute logarithmic Weil CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 4 height on P N Q ( Q ) restricts to a height function h A g : A g ( Q ) → R . Thus h A g representsthe Weil height attached to the ample line bundle obtained by restricting O (1) on P N Q to A g . Moreover, h A g takes values in [0 , ∞ ) as the Weil height function is non-negative.For M ∈ N = { , , , . . . } we write A [ M ] g for the M -fold fibered power A g × A g · · ·× A g A g over A g . Then A [ M ] g → A g is an abelian scheme.Similarly, for any morphism of schemes S → M g , the base change is C S = C g × M g S . Furthermore, the M -fold fibered power C S × S · · · × S C S is denoted by C [ M ] S . Themorphism C S → S is smooth and therefore open, thus so is C [ M ] S → S . Each fiber of thelatter morphism is a product of curves and thus irreducible. We conclude that C [ M ] S isirreducible if S is.Suppose C is a smooth curve defined over a field and A = Jac( C ). The differencemorphism C M +1 → A M determined by(2.2) ( P , P , . . . , P M ) ( P − P , . . . , P M − P ) . is well-defined; we do not need to specify a base point for the Abel–Jacobi map. It is anastonishingly powerful tool in diophantine geometry.Let us make this more precise in our relative setting. For each morphism of schemes S → M g τ −→ A g , we will construct a morphism(2.3) D M : C [ M +1] S → A [ M ] g × A g S. Indeed by the proof of [MFK94, Proposition 6.9] there is a morphism ι : C S → Pic ( C S /S )to the line bundles of degree 1. Let f , f be any two morphisms from an S -scheme T with target C S . Then the difference of ι ◦ f and ι ◦ f is a morphism T → Pic ( C S /S ) =Jac( C S ). So we get a morphism C [2] S → Jac( C S ). We compose with the natural morphismJac( C S ) → A g × A g S coming from the Torelli morphism. This construction extends to M + 1 section and yields (2.3). Fiberwise the morphism D M behaves on points as (2.2).The morphism D M in (2.3) is called the M -th Faltings–Zhang map. Note that if S is irreducible, then D M ( C [ M +1] S ) is an irreducible subvariety of the abelian scheme A [ M ] g × A g S → S .We will use the following theorem, which we proved in [DGH20], by applying [DGH20,Theorem 1.6] to [Gao, Theorem 1.2’]. Theorem 2.2.
Let S be an irreducible variety with a (not necessarily dominant) finitemorphism S → M g . Assume g ≥ and M ≥ g − . Then there exist constants c > and c ′ ≥ and a Zariski open dense subset U of D M ( C [ M +1] S ) with (2.4) ˆ h L ( P ) ≥ ch A g ( π ( P )) − c ′ for all P ∈ U ( Q ) . A strengthend height inequality.
We use the notation in the previous subsec-tion. Here we prove that the Relative Bogomolov Conjecture allows us to furthermorestrengthen the height inequality given by Theorem 2.2. Let g ≥ Proposition 2.3.
Let S be an irreducible variety with a (not necessarily dominant)finite morphism S → M g . Let M be an integer satisfying M ≥ g − if g = 2 and M ≥ g − if g ≥ . CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 5
Assume that the Relative Bogomolov Conjecture, Conjecture 1.2, holds true. Thenthere exist a constant c > and a Zariski open dense subset U of D M ( C [ M +1] S ) with (2.5) ˆ h L ( P ) ≥ c max { , h A g ( π ( P )) } for all P ∈ U ( Q ) . Proof.
The fiber of C [ M +1] S → S above s ∈ S ( Q ), is a product of M + 1 curves of genus g . The Faltings–Zhang map is generically finite on this product. So we have(2.6) codim A [ M ] g × A g S D M ( C [ M +1] S ) = M g − ( M + 1) = ( g − M − > g − ≥ dim S. Let η be the geometric generic point of S . Each fiber of D M ( C [ M +1] S ) → S is theimage of the ( M + 1)-fold power of a smooth curve under the Faltings–Zhang map. Soall fibers are irreducible and thus so is the geometric generic fiber D M ( C [ M +1] S ) η . Asa smooth curve generates its Jacobian we find that D M ( C [ M +1] S ) η is not contained in aproper algebraic subgroup of A [ M ] g × A g η . By (2.6) the image X := D M ( C [ M +1] S ) satisfiesthe assumptions of the Relative Bogomolov Conjecture. Thus there exists ǫ > X ( ǫ ; L ) Zar , the Zariski closure of X ( ǫ ; L ) = { x ∈ X ( Q ) : ˆ h L ( x ) ≤ ǫ } , is not equalto X .Let c > c ′ ≥ , and U be as in Theorem 2.2. The paragraph above implies that U \ X ( ǫ ; L ) Zar is still Zariski open and dense in X .It suffices to prove that (2.5) holds true with c a positive constant that is independentof P and with U replaced U \ X ( ǫ ; L ) Zar .Take any P ∈ ( U \ X ( ǫ ; L ) Zar )( Q ). So ˆ h L ( P ) ≥ ǫ and ˆ h L ( P ) ≥ ch A g ( π ( P )) − c ′ .We split up into the two cases depending on whether max { , h A g ( π ( P )) } ≤ max { , c ′ /c } holds or does not hold.In the first case we haveˆ h L ( P ) ≥ ǫ max { , c ′ /c } max { , h A g ( π ( P )) } and (2.5) follows with ǫ/ max { , c ′ /c } for the constant c .In the second case we have h A g ( π ( P )) > max { , c ′ /c } and hence ch A g ( π ( P )) − c ′ ≥ ch A g ( π ( P )) /
2. Thus ˆ h L ( P ) ≥ c { , h A g ( π ( P )) } . Again (2.5) holds with c/ c . (cid:3) Remark 2.4.
The proof of Proposition 2.3 is the only place where the Relative Bogo-molov Conjecture is used in the proof of Theorem 1.3.
Dichotomy on the N´eron–Tate distance between points on curves.
We usethe notation of Subsection 2.1.
Proposition 2.5.
Let S be an irreducible closed subvariety of M g .Assume that the Relative Bogomolov Conjecture, Conjecture 1.2, holds true. Thenthere exist positive constants c , c , c depending on S with the following property. Forall s ∈ S ( Q ) there is a subset Ξ s ⊆ C s ( Q ) with s ≤ c such that any P ∈ C s ( Q ) satisfies one of the following cases. CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 6 (i)
We have P ∈ Ξ s ; (ii) or (cid:8) Q ∈ C s ( Q ) : ˆ h L ( Q − P ) ≤ c − max { , h A g ( τ ( s )) } (cid:9) < c . We start the numbering of the constants from c to make the statement comparableto [DGH20, Proposition 7.1]. The proof of this proposition is similar to the proof of[DGH20, Proposition 7.1]. The main difference is that the height inequality Theorem 2.2is replaced by the strengthend version Proposition 2.3, and this allows us to remove theextra hypothesis on h A g ( τ ( s )) in [DGH20, Proposition 7.1]. Proof.
We prove the proposition by induction on dim S . The induction start dim S = 0is treated as part of the induction step.We fix M as in Proposition 2.3. Then there exist a constant c > U of X = D M ( C [ M +1] S ) satisfying the following property. For all s ∈ S ( Q )and all P, Q , . . . , Q M ∈ C s ( Q ), we have(2.7) c max { , h A g ( τ ( s )) } ≤ ˆ h ( Q − P ) + · · · + ˆ h ( Q M − P ) if ( Q − P, . . . , Q M − P ) ∈ U ( Q ) . Observe that π S ( X ) = S , where π S : A [ M ] g × M g S → S is the structure morphism. Thus S \ π S ( U ) is not Zariski dense in S . Let S , . . . , S r be the irreducible components of theZariski closure of S \ π S ( U ) in S . Then dim S j ≤ dim S − j .Note that if dim S = 0, then π S ( U ) = S and r = 0. If dim S ≥
1, then this propositionholds for all, if any, S j by the induction hypothesis. So it remains to prove the conclusionof this proposition for curves above(2.8) s ∈ S ( Q ) \ r [ j =1 S j ( Q ) . The image of s under the Torelli map is τ ( s ). For any P ∈ C s ( Q ), we will consider C s − P as a curve inside ( A g ) τ ( s ) via the Abel–Jacobi map based at P .The set W = X \ U is a Zariski closed proper subset of X . The fiber W s of W above s satisfies W s ( X s = D M ( C [ M +1] s ) as (2.8) implies s ∈ π S ( U ( Q )).We define(2.9) Ξ s = { P ∈ C s ( Q ) : ( C s − P ) M ⊆ W s } . Note that Ξ s = S Z Ξ Z where Z ranges over the irreducible components of W s andΞ Z := { P ∈ C s ( Q ) : ( C s − P ) M ⊆ Z } . We can thus apply [DGH20, Lemma 6.4] to A =( A g ) τ ( s ) , C = C s − P ⊆ A , and each Z . As Z ( D M ( C [ M +1] s ) we have Z ≤ g − g − g , is not so important; but its uniformity in s is. As W s appears as afiber of W considered as a family over S , we see that number of irreducible componentsof W s is bounded from above by a constant c ′ that is independent of s . Our estimatesimply s ≤ c where c = 84( g − c ′ is also independent of s .We have constructed Ξ s which serves as the exceptional set in part (i). We will assumethat (i) fails, i.e. , P ∈ C s ( Q ) \ Ξ s and conclude (ii). So ( C s − P ) M W s by (2.9). Ournext task is to apply [DGH20, Lemma 6.3] to C s − P and W s . CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 7
We can cover A g by a finite collection of Zariski open affine subsets V , . . . , V t . Then A g × A g V i can be embedded in P n Q × V i over V i for all i ∈ { , . . . , t } . The image of s ∈ M g ( Q ) under the Torelli morphism is τ ( s ) ∈ A g ( Q ). Assume τ ( s ) ∈ V i ( Q ) for some i . We may take C s − P as a smooth curve in P n Q . The degree of C s as a subvariety of P n Q is bounded from above independently of s . As translating inside an abelian varietydoes not affect the degree, we see that the degree of C s − P is bounded from aboveindependently of s . Recall that W s is a Zariski closed subset of X s ⊆ ( A [ M ] g ) τ ( s ) wemay identify it with a Zariski closed subset of ( P n Q ) M . The degree of W s is boundedfrom above independently of s as it is the fiber above τ ( s ) of a subvariety of ( P n Q ) M × V i .From [DGH20, Lemma 6.3] we thus obtain a number c ,i , depending only on these boundsbut not on s with the following property. Any subset Σ ⊆ C s ( Q ) with cardinality ≥ c ,i satisfies (Σ − P ) M W s ( Q ). Now let c = max( c , , . . . , c ,t ). Then c is independent of s and any subset Σ ⊆ C s ( Q ) with cardinality ≥ c satisfies (Σ − P ) M W s ( Q ).Finally set Σ = (cid:8) Q ∈ C s ( Q ) : ˆ h ( Q − P ) ≤ c − max { , h A g ( τ ( s )) } (cid:9) with c = 2 M/c .It remains to prove < c , in which case we are in case (ii) of the propositionand hence we are done. Suppose ≥ c . Then (Σ − P ) M W s ( Q ). So there exist Q , . . . , Q M ∈ Σ such that ( Q − P, . . . , Q M − P ) ∈ U ( Q ). Thus we can apply (2.7) andobtain c max { , h A g ( τ ( s )) } ≤ M c M max { , h A g ( τ ( s )) } = c { , h A g ( τ ( s )) } , a contradiction. (cid:3) Completion of the proof of Theorem 1.3 for F = Q . We follow the argumen-tation in [DGH19], or more precisely [DGH20, § C be a smooth genus g ≥ Q , and let Γ be a subgroup of Jac( C )( Q )of finite rank ρ . Let P ∈ C ( Q ).The curve C corresponds to a Q -point s c of M g, , the coarse moduli space of smoothcurves of genus g without level structure.The fine moduli space M g of smooth curves of genus g with level-4-structure admitsa finite and surjective morphism of M g, . So there exists an s ∈ M g ( Q ) that maps to s c .Thus C is isomorphic, over Q , to the fiber C s of the universal curve C g → M g above s .We thus view Γ as a finite rank subgroup of Jac( C s )( Q ), and P ∈ C s ( Q ).A standard application of R´emond’s explicit formulation of the Vojta and Mumfordinequalities [R´em00b, R´em00a] yields the following bound. There exists a constant c = c ( g ) ≥ P s ∈ C s ( Q ) such that n P ∈ C s ( Q ) : P − P s ∈ Γ , ˆ h L ( P − P s ) > c max { , h A g ( τ ( s )) } o ≤ c ρ ;we refer to Lemma 8.2 and the proof of Proposition 8.1, both [DGH20], for details.Let us start by conditionally verifying Conjecture 1.1 for P = P s . In this case itsuffices to prove(2.10) n P ∈ C s ( Q ) : P − P s ∈ Γ , ˆ h L ( P − P s ) ≤ c max { , h A g ( τ ( s )) } o ≤ c ′ ρ CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 8 for some c ′ independent of s . To do this we apply Proposition 2.5 to S = M g . Let c , c , c be from this proposition; they are independent of s . If P ∈ C s ( Q ), then P ∈ Ξ s for some Ξ s ⊆ C s ( Q ) with s ≤ c or(2.11) n Q ∈ C s ( Q ) : ˆ h L ( Q − P ) ≤ c − max { , h A g ( τ ( s )) } o < c . So to prove the desired bound (2.10) we may assume P ∈ C s ( Q ) \ Ξ s and thus (2.11).Set R = ( c max { , h A g ( τ ( s )) } ) / . We proceed by doing ball packing in the ρ -dimensional R -vector space Γ ⊗ R . It is well-known that ˆ h / defines an Euclidean norm on Γ ⊗ R .The image in Γ ⊗ R of the set in (2.10) is contained in the closed ball of radius R centeredat the image of P s . Let r ∈ (0 , R ]. By [R´em00a, Lemme 6.1] a subset of Γ ⊗ R that iscontained in a closed ball of radius R is covered by at most (1 + 2 R/r ) ρ closed balls ofradius r centered at elements of the given set. The bound in (2.11) suggests the choice r = ( c − max { , h A g ( τ ( s )) } ) / . By possibly increasing c we may assume that the quo-tient R/r = ( cc ) / lies in [1 , ∞ ). The crucial observation is that R/r is independentof s . So we can cover the image of the set in (2.10) in Γ ⊗ R with at most c ρ balls ofradius r where c ≥ s . Moreover, the balls are centered at images ofpoints P from (2.10) with P Ξ s .For such a P and by (2.11) the number of the Q ’s from C s ( Q ) that map to a singleclosed ball of radius r centered at P is less than c . Thus the number of points in (2.10)is at most c + c c ρ which is at most c ′ ρ for a suitable c ′ . This completes the proof ofthe proposition in the case P = P s .Now we turn to a general P ∈ C s ( Q ). The subgroup Γ ′ of Jac( C s )( Q ) generated by Γand P − P s has rank most ρ + 1. Now if Q ∈ C s ( Q ) − P lies in Γ, then Q + P − P s ∈ C s ( Q ) − P s lies in Γ ′ . We have just proved that the number of such Q is at most c ′ ) ≤ c ρ ≤ ( c ) ρ for a c ≥ s . (cid:3) From Q to an arbitrary base field in characteristic F = Q . Lemma 3.1.
If Conjecture 1.1 holds true for F = Q , then it holds true for an arbitraryfield F of characteristic .Proof. Without loss of generality we may and do assume that F = F .Let C , P ∈ C ( F ), and Γ be as in Conjecture 1.1 of rank ρ . By the definition of afinite rank group, there exists a finitely generated subgroup Γ of Jac( C )( F ) with rank ρ such that Γ ⊆ { x ∈ Jac( C )( F ) : [ n ] x ∈ Γ for some n ∈ N } . For each n ∈ N , define 1 n Γ := { x ∈ Jac( C )( F ) : [ n ] x ∈ Γ } . Then n Γ is again a finitely generated subgroup of Jac( C )( F ) of rank ρ . Note that { n Γ } n ∈ N is a filtered system and Γ ⊆ S n ∈ N n Γ . So in order to prove the desired bound CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 9 (1.1), it suffices to prove that there exists a constant c = c ( g ) > C ( F ) − P ) ∩ n Γ ≤ c ρ for each n ∈ N .Let n ∈ N and let γ , . . . , γ r ∈ Jac( C )( F ) be generators of n Γ such that γ ρ +1 , . . . , γ r are torsion points, we allow r to depend on n . There exists a field K n , finitely generatedover Q , such that C , P , and the γ , . . . , γ r are defined over K n . Then K n is the functionfield of some regular, irreducible quasi-projective variety V n defined over Q .Up to replacing V n by a Zariski open dense subset, C extends to a smooth family C → V n ( i.e. , C is the generic fiber of C → V n ) with each fiber being a smooth curve ofgenus g , the point P extends to a section of C → V n , and γ , . . . , γ r extend to sectionsof the relative Jacobian Jac( C /V n ) → V n . We retain the symbols P , γ , . . . , γ r for thesesections.Let v ∈ V n ( Q ). We may specialize γ , . . . , γ r at v and obtain elements γ ( v ) , . . . , γ r ( v )of the fiber Jac( C /V n ) v above v .We define the specialization of n Γ at v , which we denote with ( n Γ ) v , to be thesubgroup of Jac( C /V n ) v ( Q ) = Jac( C v )( Q ) generated by γ ( v ) , . . . , γ r ( v ). Note thatrk( n Γ ) v ≤ ρ .Suppose dim V n ≥
1, by [Mas89, Main Theorem and Scholium 1] there exists v ∈ V n ( Q )such that the specialization homomorphism n Γ → ( n Γ ) v is injective. If dim V n = 0,then V n is a point { v } and K n = Q . Here specialization is the identity and the sameconclusion holds. Thus if we denote by C v − P ( v ) the curve in Jac( C v ) = Jac( C /V n ) v obtained via the Abel–Jacobi map based at P ( v ), the specialization of P at v , then wehave C − P )( F ) ∩ n Γ ≤ (cid:0) C v − P ( v ) (cid:1) ( Q ) ∩ (cid:18) n Γ (cid:19) v . By hypothesis, Conjecture 1.1 holds true for F = Q . So the right-hand side of theinequality above has an upper bound c ρ for some c = c ( g ) ≥
1. Observe that thisbound is independent of n . Thus we have established (3.1). (cid:3) Proof of Theorem 1.3.
By hypothesis and Theorem 2.1 Conjecture 1.1 holds for F = Q .So it suffices to apply Lemma 3.1. (cid:3) Relative Bogomolov for isotrivial abelian schemes
In this section, we prove that the relative Bogomolov conjecture holds true for isotrivialabelian schemes as a consequence of S. Zhang’s Theorem [Zha98a]. An abelian scheme
A → S defined over Q is said to be isotrivial if there exists a finite and surjectivemorphism S ′ → S with S ′ irreducible such that A × S S ′ is isomorphic to A × S ′ with A an abelian variety defined over Q .Let A → S and L be as above Conjecture 1.2. Proposition 4.1.
Conjecture 1.2 holds true if
A → S is isotrivial.Proof. Let X be an irreducible subvariety of A defined over Q that dominants S suchthat X η is irreducible and not contained in any proper algebraic subgroup of A η ; here CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 10 X η means the geometric generic fiber of X and A η means the geometric generic fiber of A . Assume(4.1) codim A X > dim S. Case: Trivial abelian scheme We start by proving the proposition when
A → S is atrivial abelian scheme, i.e. , A = A × S for some abelian variety A over Q .Denote by p : A = A × S → A the natural projection. Let L be an ample andsymmetric line bundle on A defined over Q . For simplicity denote by Y = p ( X ) Zar .Then dim Y ≤ dim X , and Y is not contained in any proper algebraic subgroup of A byour assumption on X .Assume that for all ǫ >
0, the set X ( ǫ ; L ) = { x ∈ X ( Q ) : ˆ h L ( x ) ≤ ǫ } is Zariski dense in X . We will prove a contradiction to (4.1).Since L is relatively ample on A → S , there exists an integer N ≥ L ⊗ N ⊗ ( p ∗ L ) ⊗− is relatively ample on A → S . For the N´eron–Tate height functions on A ( Q ) we have N ˆ h L ≥ ˆ h p ∗ L .Take any ǫ >
0. If x ∈ X ( ǫ ; L ), then ˆ h L ( x ) ≤ ǫ , and henceˆ h L ( p ( x )) = ˆ h p ∗ L ( x ) ≤ N ˆ h L ( x ) ≤ N ǫ.
Letting x run over elements in X ( ǫ ; L ), we then obtain(4.2) p ( X ( ǫ ; L )) ⊆ Y ( N ǫ ; L ) := { y ∈ Y ( Q ) : ˆ h L ( y ) ≤ N ǫ } . We have assumed X ( ǫ ; L ) Zar = X . Applying p to both sides and taking the Zariskiclosure, we get p ( X ( ǫ ; L )) Zar = Y . Hence Y ( N ǫ ; L ) Zar = Y by (4.2). Recall that Y isnot contained in any proper algebraic subgroup of A . As N ǫ runs over all positive realnumbers, the classicial Bogomolov conjecture, proved by S. Zhang [Zha98a], implies that Y = A .So dim X ≥ dim Y = dim A = dim A − dim S , and thus codim A X ≤ dim S . Thiscontradicts (4.1). Hence we are done in this case.Case: General isotrivial abelian scheme Now we go back to an arbitrary isotrivial abelianscheme A → S .There exists a finite and surjective morphism ρ : S ′ → S , with S ′ irreducible, such thatthe base change A ′ := A × S S ′ → S ′ is a trivial abelian scheme. Denote by ρ A : A ′ → A the natural projection. Then ρ A is finite and surjective, so dim A ′ = dim A . Moreover,there is an irreducible component X ′ of ρ − A ( X ) with ρ A ( X ′ ) = X and dim X ′ = dim X .So X ′ dominates S ′ and X ′ η is irreducible, as X η is. Moreover, X ′ η is not contained in anyproper algebraic subgroup of A ′ η = A η , and codim A ′ X ′ = codim A X > dim S = dim S ′ .Finally, ρ ∗A L is relatively ample on A ′ → S ′ .We have proved the relative Bogomolov conjecture for the trivial abelian scheme A ′ → S ′ . So there exists ǫ > X ′ ( ǫ ; ρ ∗A L ) = { x ′ ∈ X ′ ( Q ) : ˆ h ρ ∗A L ( x ′ ) ≤ ǫ } is not Zariski dense in X ′ . In particular(4.3) dim X ′ ( ǫ ; ρ ∗A L ) Zar < dim X ′ = dim X. CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 11
It is not hard to check ρ A ( X ′ ( ǫ ; ρ ∗A L )) = X ( ǫ ; L ) using ˆ h ρ ∗A L ( x ′ ) = ˆ h L ( ρ A ( x ′ )) and ρ A ′ ( X ′ ) = X . Therefore, and as ρ A is a closed morphism, ρ A ( X ′ ( ǫ ; ρ ∗A L ) Zar ) = ρ A ( X ′ ( ǫ ; ρ ∗A L )) Zar = X ( ǫ ; L ) Zar . So we have dim X ( ǫ ; L ) Zar = dim ρ A ( X ′ ( ǫ ; ρ ∗A L ) Zar ) = dim X ′ ( ǫ ; ρ ∗A L ) Zar because ρ A isfinite. By (4.3) we then have dim X ( ǫ ; L ) Zar < dim X . Hence X ( ǫ ; L ) is not Zariskidense in X . We are done. (cid:3) References [ACG11] E. Arbarello, M. Cornalba, and P. Griffiths.
Geometry of Algebraic Curves, II (with a con-tribution by J. Harris) , volume 268 of
Grundlehren der mathematischen Wissenschaften .Springer-Verlag, Berlin, 2011.[CMZ18] P. Corvaja, D. Masser, and U. Zannier. Torsion hypersurfaces on abelian schemes and Betticoordinates.
Mathematische Annalen , 371(3):1013–1045, 2018.[DGH19] V. Dimitrov, Z. Gao, and P. Habegger. Uniform bound for the number of rational points on apencil of curves.
Int. Math. Res. Not. IMRN , (rnz248):https://doi.org/10.1093/imrn/rnz248,2019.[DGH20] V. Dimitrov, Z. Gao, and P. Habegger. Uniformity in Mordell–Lang for curves. arXiv:2001.10276 , 2020.[DM69] P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus.
Inst.Hautes ´Etudes Sci. Publ. Math. , (36):75–109, 1969.[DM20] L. DeMarco and N.M. Mavraki. Variation of canonical height and equidistribution.
AmericanJournal of Mathematics , 142(2):443–473, 2020.[Gao] Z. Gao. Generic rank of Betti map and unlikely intersections.
To appear in Compos. Math,arXiv: 1810.12929 .[Mas89] D. Masser. Specializations of finitely generated subgroups of abelian varieties.
Trans. Amer.Math. Soc. , 311(1):413–424, 1989.[Maz86] B. Mazur. Arithmetic on curves.
Bulletin of the American Mathematical Society , 14(2):207–259, 1986.[MFK94] D. Mumford, J. Fogarty, and F. Kirwan.
Geometric invariant theory , volume 34 of
Ergebnisseder Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] .Springer-Verlag, Berlin, third edition, 1994.[MZ12] D. Masser and U. Zannier. Torsion points on families of squares of elliptic curves.
Mathema-tische Annalen , 352(2):453–484, 2012.[MZ14] D. Masser and U. Zannier. Torsion points on families of products of elliptic curves.
Advancesin Mathematics , 259:116 – 133, 2014.[MZ15] D. Masser and U. Zannier. Torsion points on families of simple abelian surfaces and Pell’sequation over polynomial rings (with an appendix by E. V. Flynn).
Journal of the EuropeanMathematical Society , 17:2379–2416, 2015.[MZ18] D. Masser and U. Zannier. Torsion points, Pell’s equation, and integration in elementaryterms. preprint, submitted , 2018.[OS80] F. Oort and J. Steenbrink. The local Torelli problem for algebraic curves. In
Journ´ees deG´eometrie Alg´ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 , pages 157–204. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980.[Pin89] R. Pink.
Arithmetical compactification of mixed Shimura varieties . PhD thesis, Bonner Math-ematische Schriften, 1989.[Pin05] R. Pink. A Common Generalization of the Conjectures of Andr´e-Oort, Manin-Mumford, andMordell-Lang.
Preprint , page 13pp, 2005.[R´em00a] G. R´emond. D´ecompte dans une conjecture de Lang.
Invent. Math. , 142(3):513–545, 2000.[R´em00b] G. R´emond. In´egalit´e de Vojta en dimension sup´erieure.
Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4) , 29(1):101–151, 2000.
CONSEQUENCE OF THE RELATIVE BOGOMOLOV CONJECTURE 12 [Ull98] E. Ullmo. Positivit´e et discr`etion des points alg´ebriques des courbes.
Ann. of Math. (2) ,147(1):167–179, 1998.[Zan12] U. Zannier.
Some problems of unlikely intersections in arithmetic and geometry , volume 181of
Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2012. Withappendixes by David Masser.[Zha98a] S. Zhang. Equidistribution of small points on abelian varieties.
Ann. of Math. (2) , 147(1):159–165, 1998.[Zha98b] S Zhang. Small points and Arakelov theory. In
Proceedings of the International Congress ofMathematicians. Volume II , pages 217–225, 1998.
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