Unlikely intersections between isogeny orbits and curves

Fix an abelian variety $A_0$ and a non-isotrivial abelian scheme over a smooth irreducible curve, both defined over the algebraic numbers. Consider the union of all images of translates of a fixed finite-rank subgroup of $A_0$, also defined over the algebraic numbers, by abelian subvarieties of $A_0$ of codimension at least $k$ under all isogenies between $A_0$ and some fiber of the abelian scheme. We characterize the curves inside the abelian scheme which are defined over the algebraic numbers, dominate the base curve and potentially intersect this set in infinitely many points. Our proof follows the Pila-Zannier strategy.


Introduction
Let K be a field of characteristic zero, let S be a geometrically irreducible smooth curve and let A → S be an abelian scheme over S of relative dimension g, both defined over K. The structural morphism will be denoted by π : A → S and is smooth and proper. For any (possibly non-closed) point s of S and any subvariety V of A, we denote the fiber of V over s by V s . The zero section S → A is denoted by .
We fix an algebraic closureK of K. All varieties that we consider will be defined overK, unless explicitly stated otherwise. All varieties will be identified with the set of their closed points over a prescribed algebraic closure of their field of definition. Subvarieties will always be closed. By "irreducible", we will always mean "geometrically irreducible". If F is any field extension of the field over which the variety V is defined, we will denote the set of points of V that are defined over F by V (F ). If A is an abelian variety, we denote by A tors the set of its torsion points.
We fix an abelian variety A 0 of dimension g and a finite set of Z-linearly independent points γ 1 , . . . , γ r in A 0 . The set can also be empty (i.e. r = 0). We define Γ = {γ ∈ A 0 ; ∃N ∈ N: N γ ∈ Zγ 1 + . . . + Zγ r }, a subgroup of A 0 of finite rank (and every subgroup of A 0 of finite rank is contained in a group of this form), for us N = {1, 2, 3, . . .}.
The (g − k)-enlarged isogeny orbit of Γ (in the family A) is defined as Γ = {p ∈ A s ; s ∈ S, ∃φ : A 0 → A s isogeny and an abelian subvariety B 0 ⊂ A 0 of codimension ≥ k such that p ∈ φ(Γ + B 0 )}. (1.1) This condition is equivalent to the existence of an isogeny ψ : A s → A 0 with ψ(p) ∈ Γ + B 0 . The isogeny orbit of Γ is defined as Γ . Let ξ be the generic point of S. We fix an algebraic closure K(S) of K(S) and let A K(S)/K ξ , Tr denote the K(S)/K-trace of A ξ , as defined in Chapter VIII, §3 of [25], where we consider A ξ as a variety over K(S) by abuse of notation. We call A isotrivial if Tr A K(S)/K ξ = A ξ . In this article, we investigate the following conjecture, a slightly modified version of Gao's Conjecture 1.2, which he calls the André-Pink-Zannier conjecture, in [16]. We need to formulate the conclusion in this somewhat involved manner in order to account for the fact that there can exist abelian subvarieties of A ξ and points in (A ξ ) tors that are not defined overK(S) and that the morphism Tr isn't necessarily defined overK(S). It can be considered one relative version of the Mordell-Lang conjecture, proven for abelian varieties by Vojta [59], Faltings [12] and Hindry [21] and in its most general form by McQuillan in [33], in analogy to the relative Manin-Mumford results proven by Masser and Zannier in e.g. [32]. As we can always assume that K is finitely generated over Q and then embed it in C, it suffices to prove the conjecture for subfields of C.
Prima facie, Gao's conjecture only concerns irreducible subvarieties of the universal family of principally polarized abelian varieties of fixed dimension and fixed sufficiently large level structure. However, we can assume without loss of generality that A is contained in a suitable universal family A g,l corresponding to principally polarized abelian varieties of dimension g with so-called orthogonal level l-structure (cf. Sections 2 and 8), which reduces Conjecture 1.1 to the case considered by Gao. The condition that the base S in this situation is a weakly special curve in the moduli space seems to be missing in our formulation of the conjecture, but it follows directly from Orr's Theorem 1.2 in [37] that Conjecture 1.1 can be further reduced to this case. The conjecture is stronger than Gao's in that it involves a subgroup of rank possibly larger than 1 and doesn't demand that the isogenies are polarized. It is weaker in that the base variety S is assumed to be a curve.
Gao showed in Section 8 of [16] that Conjecture 1.1 follows from Pink's Conjecture 1.6 in [46] in the more general setting of generalized Hecke orbits in mixed Shimura varieties, where it is enough to assume Pink's conjecture for all fibered powers of universal families of principally polarized abelian varieties of fixed dimension and fixed, sufficiently large level structure. By Theorem 3.3 in [47], Conjecture 1.6 in [46] is a consequence of Pink's even more general Conjecture 1.1 in [47] on unlikely intersections in mixed Shimura varieties. If Γ has rank zero, Conjecture 1.1 is contained in a special-point conjecture of Zannier (see [16], Conjecture 1.4).
Progress towards Conjecture 1.1 has only been made if V = C is a curve or if the rank of Γ is zero. Furthermore, many results are confined to the case where K is a number field. Lin and Wang have proved the conjecture for K a number field, V a curve, Γ finitely generated and A 0 simple (Theorem 1.1 in [26]). Habegger has proved it for K a number field, Γ of rank zero and A a fibered power of a non-isotrivial elliptic scheme (Theorem 1.2 in [19]). Pila has proved it for arbitrary K, Γ of rank zero and A inside a product of elliptic modular surfaces (Theorem 6.2 in [40]). Gao has proved it for arbitrary K and Γ of rank zero (Theorem 1.5 in [16]) as well as for arbitrary K, V a curve and Γ of rank at most one, but in this case he has to fix polarizations of A 0 and A and assume that the isogenies are polarized (Theorem 1.6 in [16]).
From now on, we will always assume that K ⊂ C is a number field and take asK =Q its algebraic closure in C. We expect however that Theorem 1.3 can be generalized to the transcendental case in the same way as Gao's by use of the Moriwaki height instead of the Weil height together with specialization arguments.
The purpose of this paper is twofold: First, we prove Conjecture 1.1 in Theorem 1.3 if K is a number field and V = C is a curve. Second, we investigate what happens when C ∩ A [k] Γ is infinite for some arbitrary k ∈ {0, . . . , g}. Here, the case k = g corresponds to Conjecture 1.1. If k < g, the condition is weaker (if k = 0, it is void), so we expect a weaker conclusion. We prove the strongest possible conclusion in Theorem 1.2, of which Theorem 1.3 thus becomes a special case.
The problem of intersecting a fixed subvariety with algebraic subgroups originates in works of Bombieri-Masser-Zannier [7] and Zilber [61] for powers of the multiplicative group. The analogous problem in a fixed abelian variety has also been the object of much study; we just mention the work of Habegger and Pila [20], from which we use several results in our proof. The intersection of a subvariety of a fixed abelian variety with translates of abelian subvarieties by points of a subgroup of finite rank has been studied by Rémond in e.g. [51]. While there has been intensive study of unlikely intersections between a curve in an abelian scheme and flat algebraic subgroup schemes, culminating in the article by Barroero and Capuano [4], ours seems to be the first result that combines intersecting with positive-dimensional algebraic subgroups with an isogeny condition on the fiber.
We can now state our main results. Recall that S is a smooth irreducible curve and A → S is an abelian scheme, both defined over K, while C ⊂ A is a closed irreducible curve, defined overQ, A 0 is an abelian variety defined overQ, γ 1 , . . . , γ r ∈ A 0 (Q) and Γ ⊂ A 0 is the subgroup of all γ ∈ A 0 such that N γ ∈ Zγ 1 + . . . + Zγ r for some N ∈ N.
Γ ∩ C is infinite and π(C) = S, then C is contained in an irreducible subvariety W of A of codimension ≥ k with the following property: OverQ(S), every irreducible component of W ξ is a translate of an abelian subvariety of A ξ by a point in Theorem 1.3. Suppose that A → S is not isotrivial. If A Γ ∩ C is infinite, then one of the following two conditions is satisfied: (i) The curve C is a translate of an abelian subvariety of A s by a point of A Γ ∩ A s for some s ∈ S.
(ii) The zero-dimensional variety C ξ is contained in (A ξ ) tors +Tr AQ From Theorem 1.3, we can deduce the following corollary: Corollary 1.4. Let A g,l be the moduli space of principally polarized abelian varieties of dimension g with orthogonal level l-structure as defined in Section 2 and l sufficiently large and let A and B be abelian varieties with dim B = g. Let C ⊂ A g,l × A be a closed irreducible curve and let pr 1 : C → A g,l and pr 2 : C → A be the canonical projections. Let Γ ⊂ A be a subgroup of finite rank and let Σ ⊂ A g,l be the set of s ∈ A g,l corresponding to abelian varieties that are isogenous to B. If C ∩ (Σ × Γ ) is infinite, then either pr 1 or pr 2 is constant.
We thereby prove Conjecture 1.7 of Buium and Poonen in [9]: If S is a modular curve or a Shimura curve, then a Zariski open subset S of S has a moduli interpretation which yields a quasi-finite forgetful modular morphism from S to the coarse moduli space A g of principally polarized abelian varieties of dimension g ∈ {1, 2}. Similarly, we have a quasi-finite morphism A g,l → A g . We can then form the curve S × Ag A g,l , which admits quasi-finite morphisms to S and A g,l , and reduce the conjecture to Corollary 1.4. The conjecture of Buium and Poonen has been proven independently by Baldi in [3] through the use of equidistribution results. He was also able to replace Γ by a fattening Γ for some > 0 (see [3] for the definition of Γ ). Such an extension seems to lie outside the reach of our methods though.
The proof of Theorem 1.2 uses point counting and o-minimality and in particular a later refinement of the theorem of Pila-Wilkie on rational points on definable sets in [43]. In applying this result to problems of unlikely intersections in diophantine geometry, we follow the standard strategy as devised by Zannier for the new proof of the Manin-Mumford conjecture by Pila and him in [44]. It is described in Zannier's book [60]. In Section 2, we introduce some notation and make several reduction steps.
In Sections 3 and 4, we bound the "height" of all important quantities from above in terms of the degree of the varying point p = φ(q) ∈ A [k] Γ ∩ C over the fixed number field K. The main new ideas of this article are to be found in these two sections. In order to treat non-polarized isogenies, we extend a result by Orr to show that the isogeny φ between A 0 and A s can be chosen such that certain associated quantities are bounded in the required way -first of all, we apply the isogeny theorem of Masser-Wüstholz to show that the degree of the isogeny can be bounded in this way. As a consequence of our extension of Orr's result we can then bound the height of q for this choice of φ. (After maybe enlarging Γ, we can fix for each s ∈ S such that A 0 and A s are isogenous one choice of isogeny -see Lemma 2.2.) We bound the degree of the smallest translate of an abelian subvariety of A r+1 0 by a torsion point that contains (q, γ 1 , . . . , γ r ) through an application of a proposition by Habegger and Pila. Using this and a lemma of Rémond, we can then write q = γ + b with γ ∈ Γ of controlled height and b in an abelian subvariety of controlled codimension and degree. If N is the smallest natural number such that N γ ∈ r i=1 Zγ i , we finally bound N by applying a lemma of Habegger and Pila, some elementary diophantine approximation and lower height bounds on abelian varieties due to Masser. In Section 5, we give a brief introduction to o-minimal structures in as much depth as is necessary to state a variant of the Pila-Wilkie theorem, due to Habegger and Pila, on "semirational" points of bounded height.
In Section 6, the definability in a suitable o-minimal structure of the analytic uniformization map associated to our abelian scheme is shown, when restricted to some fundamental domain, by use of a theorem of Peterzil-Starchenko. In Section 7, we record the necessary algebraic independence result of "logarithmic Ax" type by Gao, which generalizes work by André in [2] and by Bertrand in [5].
Finally, we put all the pieces together in Section 8 and prove Theorem 1.2, Theorem 1.3 and Corollary 1.4.

Preliminaries and Notation
For a rational number α = a b with a ∈ Z, b ∈ N and gcd(a, b) = 1, we define its affine height H(α) = max{|a|, |b|}. We will fix once and for all a square root of −1 inside C that we denote by √ −1 -this yields maps Re : C → R and Im : C → R in the usual way. For an integral domain R, we denote the space of m×n-matrices with entries in R by M m×n (R). We write M n (R) for M n×n (R). For a matrix A = (a ij ) ∈ M n (Q), we define its height H(A) = max i,j H(a ij ). The complex conjugate of a matrix A with complex entries will be denoted by A and the transpose by A t . The n-dimensional identity matrix will be denoted by E n . The row-sum norm of a matrix A ∈ M m×n (C) will be denoted by A . For a vector v = (v 1 , . . . , v n ) t ∈ C n , we will write v for max j=1,...,n |v j |. Note that Av v for all A ∈ M m×n (C). Vectors will always be column vectors. By applying Re and Im to each entry, we obtain maps from M n (C) to M n (R) that by abuse of language will also be called Re and Im.
If A is an arbitrary abelian variety over an arbitrary field, we denote its dual abelian variety byÂ. If φ : A → B is an isogeny, the dual isogeny will be denoted byφ :B →Â.
For our proof of Theorem 1.3, we will restrict ourselves in the following sections to subfamilies of the universal family A g,l → A g,l of principally polarized abelian varieties with so-called orthogonal level l-structure for a natural number l ≥ 16 which is divisible by 8 and a perfect square and identify π and with the natural projection and zero section of that family. If H g denotes the Siegel upper half space in dimension g (i.e. symmetric matrices in M g (C) with positive definite imaginary part), then A g,l is a quotient of H g × C g by the semidirect product of the congruence subgroup of Sp 2g (Z) with Z 2g , where diag denotes the diagonal of a matrix. We will show at the end of our work in Section 8 how to deduce the result for arbitrary families. The group G(l, 2l) is the same as the group Γ(l, 2l) defined on p. 422 of [30]. Let G lEg (lE g ) 0 be defined as in Section 8.9 of [6]. Then there is an isomorphism from G(l, 2l) to G lEg (lE g ) 0 given by sending M to E g 0 0 l −1 E g M E g 0 0 lE g -see [6], Section 8.8 and 8.9, and note that l is even.
The group law on the semidirect product is given by (M , z )(M, z) = (M M, z + (M ) −t z) and the action of the group is given by The action of course extends to an action of Sp 2g (R) R 2g (with the same group law) and then also restricts to the usual action of Sp 2g (R) on H g . If M ∈ Sp 2g (R) and τ ∈ H g , we will denote this last action as above by M [τ ] to avoid confusion with ordinary matrix multiplication. By applying Proposition 8.2.5 in [6] and Cartan's Exposé 11 in Volume 2 of [1], we see that our universal family is a complex analytic space because the group action is proper and discontinuous -Proposition 8.2.5 of [6] only says that the action of G(l, 2l) on H g is proper and discontinuous, but this quickly implies the same for the action of its semidirect product with Z 2g on H g × C g . However, the universal family is in fact a quasi-projective variety, defined over Q. In the following proposition, we recall some well-known facts about it.
Proposition 2.1. There exist holomorphic maps and ι : H g → P l g −1 (C) with the following properties: (i) There is a commutative diagram where the vertical maps are projections to the first factor. (ii) We have exp(τ, z) = exp(τ , z ) if and only if (τ, z), (τ , z ) lie in the same G(l, 2l) Z 2g -orbit and exp descends to an analytic embedding of the quotient. Similarly, we have ι(τ ) = ι(τ ) if and only if τ , τ lie in the same G(l, 2l)-orbit and ι descends to an analytic embedding of the quotient. (iii) The images exp(H g × C g ) and ι(H g ) are locally closed with respect to the Zariski topology in P l g −1 (C)×P l g −1 (C) and P l g −1 (C) respectively. They are irreducible smooth varieties, defined over Q. (iv) exp(H g × C g ) → ι(H g ) is an abelian scheme, defined over Q, with zero section p → (p, p). where (vi) The very ample line bundle on exp({τ } × C g ) that is induced by this embedding is the l-th tensor power of a symmetric ample line bundle. Under the uniformization exp({τ } × C g ) C g /Ω τ Z 2g given by exp, the Hermitian form on C g induced by this second line bundle is given by the matrix (Im τ ) −1 .
Proof. We can explicitly give the maps, using the classical theta functions. For this, we define for τ ∈ H g , z ∈ C g and a, b ∈ Q g . For c ∈ Q g and (τ, z) ∈ H g × C g , we put We then define φ(τ, z) = [θ c 0 (τ, z) : . . . : θ c l g −1 (τ, z)] and ι(τ ) = φ(lτ, 0) as well as exp(τ, z) = (φ(lτ, 0), φ(lτ, lz)), where the c i run over the set {0, 1 l , . . . , 1 − 1 l } g (i = 0, . . . , l g − 1). Property (i) now follows directly from the definitions. For property (ii), we refer to Chapter 8 of [6] and Chapter V of [23]. Note that due to the above-mentioned isomorphism between G lEg (lE g ) 0 and G(l, 2l) τ and τ lie in the same G(l, 2l)-orbit if and only if lτ and lτ lie in the same G lEg (lE g ) 0orbit. It follows from [34], §1, that the actions are free, so the quotient maps are covering maps. The map ι descends to a proper map from the quotient to its image, since over every point of its image lies exactly one point of the normalization of the closure of the image by [24]. One can use Theorem 4.5.1 of [6] to show that not only ι, but also exp descends to an analytic embedding.
For properties (iii) and (iv), see Section 3 of [30] and the references given there, in particular [35]. Smoothness and irreducibility follow from the fact that the quotients are connected complex analytic manifolds. Property (v) follows from (ii) and the choice of zero section.
Property (vi) follows by computing the factor of automorphy of the embedding exp(τ, ·) of C g /Ω τ Z 2g : An elementary computation shows that for all c ∈ {0, 1 l , . . . , 1 − 1 l } g and all m, n ∈ Z g . By Remark 8.5.3(d) in [6], this factor of automorphy belongs to the l-th tensor power of a symmetric ample line bundle that under the given uniformization is associated to the Hermitian form given by (Im τ ) −1 on C g .
Using the proposition, we may identify exp(H g × C g ) and ι(H g ) with A g,l (C) and A g,l (C) and use A g,l and A g,l for the corresponding quasiprojective varieties, defined over Q. We will denote the Zariski closures in P l g −1 and P l g −1 × P l g −1 of these varieties by A g,l and A g,l respectively; these are (usually highly singular) projective varieties, also defined over Q. The projection from P l g −1 × P l g −1 onto the first factor yields a morphism π : A g,l → A g,l . The embedding from the proposition yields very ample line bundles L on A g,l and L on A g,l .
From now on, we assume that S ⊂ A g,l is an irreducible, smooth, locally closed curve (not necessarily closed in A g,l ), A = π −1 (S) and C ⊂ A is an irreducible closed curve. We denote by C and S the Zariski closures of C and S in A g,l and A g,l respectively. The abelian scheme A → S and the curve S are defined over K.
After maybe enlarging K, we can and will assume without loss of generality that A 0 , the addition morphism A 0 × A 0 → A 0 , the inversion morphism A 0 → A 0 , C and C are defined over K and that A 0 is principally polarized. For this, we might have to replace A 0 by an isogenous abelian variety and Γ by its pre-image under the corresponding isogeny. This doesn't change the isogeny orbit, so doesn't change the statement we want to prove.
We fix a symmetric ample line bundle L 0 which gives us a principal polarization on A 0 and fix once and for all a uniformization C g /Ω τ 0 Z 2g of A 0 (C) such that the Hermitian form on C g associated to L 0 is given by (Im τ 0 ) −1 , Ω τ 0 = ( τ 0 Eg ) and τ 0 lies in the Siegel fundamental domain (see Definition 3.2). We denote the corresponding map C g → A 0 (C) by exp 0 . Using Weil's Height Machine (see [22], Theorem B.3.2 and B.3.6), we also get a (logarithmic projective) height h A 0 = h A 0 ,L 0 on A 0 . With the usual construction due to Néron and Tate (see [22], Theorem B.5.1) we then obtain a canonical height h A 0 on A 0 .
After maybe enlarging K again, we can assume that L 0 is defined over K, γ 1 , . . . , γ r ∈ A 0 (K), and every endomorphism of A 0 is defined over K.
Since the endomorphism ring of A 0 is finitely generated as a Z-module, we may assume that Γ is mapped into itself by every endomorphism of A 0 by enlarging Γ if necessary (which only makes Theorems 1.2 and 1.3 stronger). We will generally assume that r ≥ 1 for simplicity -one can either ensure this by enlarging Γ and K or one can check that our proof also works mutatis mutandis if r = 0.
The line bundle L restricts to a very ample line bundle L S on S. For each s ∈ S, the restriction of L to A s is a very ample symmetric line bundle L s by Proposition 2.1(vi). From the embeddings into projective space by theta functions, we directly obtain associated heights h S on S and h s on A s (s ∈ S) as well as a canonical height h s on A s .
The following technical lemma shows that for each s ∈ S we can fix an isogeny φ s in the definition of A Proof. We prove the non-trivial inclusion "⊂". Suppose that p ∈ A Γ . Then p lies in some A s (s ∈ S) such that A s and A 0 are isogenous. By definition, there is an isogeny φ : A 0 → A s , an abelian subvariety B 0 of A 0 of codimension ≥ k and γ ∈ Γ such that p ∈ φ(γ + B 0 ).
We take φ s as an isogeny between A s and A 0 of minimal degree, i.e. there exists no isogeny ψ : A 0 → A s of degree less than deg φ s . By Théorème 1.4 of Gaudron-Rémond in [18], which improves a theorem of Masser-Wüstholz ( [29], p. 460), there exist constants c M W and κ M W , depending only on independently of s. Note that A s and A 0 are both defined over K(s).

Height bounds for isogenies
In the previous section, we took as φ s just any isogeny between A 0 and A s of minimal degree. This is fine in the case of elliptic curves, but in arbitrary dimension, we have to pick the distinguished isogeny more carefully. This will be achieved in Proposition 3.3 and Corollary 3.4, where we replace φ s by φ s • σ for some well-chosen automorphism σ of A 0 . Proposition 3.3(ii) and Corollary 3.4(ii) are essentially contained in Orr's work [37], albeit formulated rather differently, and our proofs of these results basically run along the same lines as his. Another way to get the desired bounds on quantities associated to an isogeny between A 0 and A s (s ∈ S) would be to replace the use of Orr's Proposition 4.2 from [37] with the endomorphism estimate from Lemma 5.1 of Masser and Wüstholz in [31] for A 0 × A s (an improved, completely explicit bound can be deduced from Section 9 of [18], Lemme 2.11 of [49] and Minkowski's second theorem) and an argument as in Section 6 of [31]. Afterwards, one could continue as we do here and obtain bounds that are polynomial (in the sense of (3.1)) not necessarily in the degree of the isogeny, but certainly in [K(s) : K].
Before we can prove the proposition, we need the following technical lemma.
Proof. Using elementary row operations from GL 2g (Z), we can write M = M 1 P 1 with M 1 ∈ GL 2g (Z) and P 1 ∈ M 2g (Z) upper triangular. The (nonzero) diagonal entries of P 1 are then bounded by | det M | and after more row operations we can assume that the entries above the diagonal entry d lie in the set {0, 1, . . . , |d| − 1}. So we can assume without loss of generality that H(P 1 ) is bounded by | det M |, which is of course polynomially bounded in H. Then is also polynomially bounded in H, so it suffices to prove the lemma for M 1 and H instead of M and H. The lemma is now a consequence of Orr's Lemma 4.3 in [37], which can be reformulated as asserting that there exists P 2 ∈ GL 2g (Z) of height bounded polynomially in H such that M 1 P 2 ∈ Sp 2g (Z).
Before we can state the next theorem, we have to define what a Siegel fundamental domain for the action of (a finite-index subgroup of) Sp 2g (Z) on H g is. We give the definition that goes back to Siegel in [53], §2.

Definition 3.2.
(1) A positive definite symmetric matrix is a subgroup of finite index and g 1 = E 2g , g 2 , . . . , g n is a system of representatives for its right cosets, then n j=1 g j F is called a Siegel fundamental domain for G.
It is a classical fact that for only finitely many M ∈ Sp 2g (Z) there exists some τ in the Siegel fundamental domain with M [τ ] also in the Siegel fundamental domain and that every element of H g can be brought into the Siegel fundamental domain by some element of Sp 2g (Z). The same facts then easily follow for every Siegel fundamental domain for some subgroup of Sp 2g (Z) of finite index. This is everything we will need to know about Siegel fundamental domains in this section. Proposition 3.3. Let A and B be two abelian varieties of dimension g, defined over C and uniformized as Let φ : A → B be an isogeny. Then there exist constants C and κ, depending only on F , A, Ω A and M, but not on B or φ, a natural number n ∈ N, an automorphism σ : A → A and a matrix Φ ∈ M 2g (Z) such that (ii) Φ is the rational representation of φ • σ with respect to the lattice bases given by Ω A and Ω B and H(Φ) ≤ C(deg φ) κ .
Proof. Let φ M and φ N be the principal polarizations induced by M and N respectively.
It is also totally positive (or positive definite in the terminology of [37]) by Theorem Therefore, we can apply Orr's Proposition 4.2 in [37] and deduce that there is a constant c, depending only on A and Ω A , and σ ∈ Aut(A) such that the rational representation of σ • ψ • σ with respect to the lattice given by Ω A has height bounded by c(deg φ) 2 (we choose (End A, ) as (R, †) and the rational representation with respect to the lattice given by Ω A as ρ). We , so we can replace φ by φ • σ and ψ by σ • ψ • σ and verify (i) and (ii) for this new φ (and σ = id), where Φ ∈ M 2g (Z) is the rational representation of φ with respect to the lattice bases given by Ω A and Ω B . We have | det Φ| = deg φ = 0.
Let H M and H N be the Hermitian forms on C g associated to M and N respectively and let A and B be the matrices in M 2g (R) that represent the symmetric positive definite forms Re H M and Re H N with respect to the lattice bases given by Ω A and Ω B respectively. Let M 1 ∈ M 2g (Z) be the rational representation of ψ with respect to the lattice basis given by By taking the analytic representations of both sides, where the dual abelian varieties are canonically uniformized as quotients of the vector space of C-antilinear maps from C g to C, it follows (with Lemma 2.4.5 from [6]) that for all v, w ∈ C g , where we use φ and ψ also for the linear maps from C g to C g corresponding to the analytic representations of φ and ψ with respect to the given uniformization. By taking real parts and passing to rational representations, we deduce that Let H φ * N be the Hermitian form associated to φ * N . The ampleness of (φ * N ) ⊗n ⊗ M ⊗(−1) is equivalent to the positive definiteness of its Hermitian form H n = nH φ * N −H M and this is equivalent to the positive definiteness of the symmetric bilinear form Re H n . One computes that Re H n is represented by M 2 = nΦ t B Φ − A with respect to the lattice given by Ω A . Let v ∈ R 2g be an arbitrary non-zero vector and 1 v), and using the Cauchy-Schwarz inequality for the scalar product given by In order to make this quantity positive, n must be bigger than the operator norm of M −1 1 with respect to the scalar product given by M 3 , i.e. Since B and hence M 3 is symmetric and positive definite, there is a −t , so the coefficients ofM 3 andM 3 −1 must be similarly bounded.
and obtain a bound of the desired form. This proves (i).
For (ii), we have Φ t [T B ] = T A for the partial action of GL 2g (Q) on H g that restricts to the usual action of Sp 2g (Z).
By Lemma 3.1, we can write Φ = SP , where S ∈ Sp 2g (Z) and P ∈ Φ represents the imaginary part of the Hermitian form H φ * N with respect to the lattice basis given by Ω A (here we use that the lattice basis associated to Ω B is symplectic with respect to H N ). We have Furthermore, we know that Siegel's definition from [54] is used, which demands that (Im τ ) −1 instead of Im τ is Minkowski-reduced, since by Lemma 3.3 of [41] and Lemma 3.1(3) of [42] one can switch between the two fundamental domains in a (polynomially) controlled way.
In order to bound the absolute values of the coefficients of S t [T B ] as well as (det Recall that det P = det Φ = 0. As we have a bound on the coefficients of M 3 and on H(P ), we deduce a similar bound for the coefficients of M 4 . If , then we see that M 4 represents the real part of H N with respect to the lattice basis given by the columns of ( T B S t . In order to compute M 4 , it is useful to choose the basis given by the columns of T B S t 3 + S t 4 for C g . That this matrix has non-zero determinant (and hence its columns form a basis) follows from the proof that Sp 2g (Z) acts on H g by (U, τ ) → U [τ ].
With respect to this new basis of C g , the lattice basis given by the columns of ( T B S t is given by the matrix ( S t [T B ] Eg ). Furthermore, the Hermitian form H N is given by ( ) −1 with respect to this new basis of C g (see the calculation in [6], p. 214).
With this new basis for C g , it is easy to compute Here In order to state the next corollary, we introduce the following notation that will also be used in the following sections: We write f g for (positive) quantities f and g, if there exist constants c > 0 and κ > 0, depending on K, A 0 , L 0 , τ 0 , Γ, l, A, L, S, C and the choice of a Siegel fundamental domain for G(l, 2l) such that f ≤ c max{1, g} κ .
(3.1) The choice of a Siegel fundamental domain for G(l, 2l) will be made implicitly in Proposition 6.1.
and thus we may take M = n. Note that G(l, 2l) has finite index in Sp 2g (Z) and that τ 0 was already chosen in the Siegel fundamental domain for Sp 2g (Z). The implicit constants depend only on A 0 , L 0 , τ 0 and the chosen Siegel fundamental domain, but are independent of s and τ .
Finally, we record a lemma due to Rémond that allows us to bound the height of a basis of the lattice corresponding to an abelian subvariety of A 0 in terms of the degree of the abelian subvariety.
Lemma 3.5. Let B 0 be an abelian subvariety of A 0 of codimension k and denote by deg B 0 its degree with respect to the ample line bundle L 0 . Under the identification of R 2g with C g given by u → Ω τ 0 u, there exists a matrix H ∈ M 2g×2(g−k) (Z) such that exp −1 0 (B 0 (C)) = {Hy + z; y ∈ R 2(g−k) , z ∈ Z 2g }, Ω τ 0 H has rank equal to g − k and H deg B 0 . Here, exp 0 and Ω τ 0 are defined as in Section 2.
Proof. We follow Rémond's construction in Section 4 of [50]. We obtain a basis ) of the connected component of exp −1 0 (B 0 (C)) containing 0 under the given identification of R 2g and C g . Here, v 1 , . . . , v 2g is a suitable basis of Z 2g that is chosen depending on L 0 , but independently of B 0 , and the λ (i) j are integers. By an inequality on p. 531 of [50], we have where · is a Euclidean norm on R 2g induced by L 0 . This norm is bounded from below on Z 2g \{0} by a positive constant that doesn't depend on B 0 , which implies that ).
Since all norms on finite-dimensional real vector spaces are equivalent and · doesn't depend on B 0 , it follows that |λ (i) j | deg B 0 (i = 1, . . . , 2(g − k), j = 1, . . . , 2g). We deduce that the coordinates of w 1 , . . . , w 2(g−k) with respect to the basis v 1 , . . . , v 2g of Z 2g (which is not necessarily the standard one) are bounded. However, this basis is chosen independently of B 0 and so we obtain a comparable bound for the coordinates with respect to the standard basis.
We now take as H the matrix with columns w 1 , . . . , w 2(g−k) . The columns of the matrix Ω τ 0 H span the connected component of exp −1 0 (B 0 (C)) containing 0 seen as a (g − k)-dimensional vector subspace of C g and so this matrix has rank equal to g − k.

Galois orbit bounds
In this section, we show that virtually all occurring important quantities can be bounded polynomially in terms of [K(p) : K], where p is a point in A [k] Γ ∩ C (reversing the direction of the inequalities leads to lower bounds for [K(p) : K] in terms of these other quantities -hence the title "Galois orbit bounds"). We will need two lemmata before we can prove the crucial Proposition 4.3. From now on, we will always take the isogeny given by Corollary 3.4 as φ s . There might be some ambiguity in the choice of τ if it lies on the boundary of the Siegel fundamental domain for G(l, 2l), but this ambiguity doesn't change the construction in Proposition 3.3 -which only depends on the principal polarization induced by L s and the data associated to A 0 -and hence has no influence on φ s . Likewise, the implicit constants in the estimates are the same for any choice of τ in the Siegel fundamental domain. Proof. We will use c 1 , c 2 , . . . for constants depending on K and A 0 , but independent of s. We will denote the stable Faltings height of A s as defined in [11] by h F (A s ). By Faltings' Lemma 5 in [11], we have (4.1)

By an inequality of Bost-David (Pazuki's Corollary 1.3 (1) in [38]), we know that
for some constants c 3 and c 4 , depending only on g and l. Our choice of embedding of A g,l and A g,l into projective space through the use of Theta functions means that our h S (s) differs from the Theta height of A s in Pazuki's work with l = r 2 only by an amount that is bounded independently of s: Pazuki uses another norm at the archimedean places for the definition of his height and he uses another coordinate system as he notes after his Definition 2.6, but by [23], p. 171, this coordinate system is related to ours by an invertible linear transformation with algebraic coefficients. We deduce that h S (s) ≤ c 5 max{h F (A s ), 1}.  Let p ∈ C with s = π(p) ∈ S and suppose that C is not contained in A s . Then we have h s (p) h S (s).
Our proof even yields a bound that is linear in h S (s), but a polynomial bound will suffice for our purposes. We note that it is crucial for this lemma that C is a curve and not a subvariety of A of higher dimension. Indeed, the main obstacle that one encounters attempting to generalize Theorem 1.3 to higher-dimensional subvarieties V ⊂ A which dominate the base is the lack of such a height bound for (a large enough subset of) the points in A Γ ∩ V.
Proof. We use c 6 , . . . for constants that depend only on A and C. Let for the moment s ∈ S and p ∈ A s be arbitrary. We will first bound h s (p) in terms of h s (p) and h S (s). It would be possible to use Silverman's Theorem A in [56] for this; there is however the problem that A g,l and A g,l are usually not smooth, so one would either need to construct a more sophisticated (i.e. smooth) compactification of the universal family (this was achieved by Pink in his dissertation [45]) or adapt Silverman's proof by using Cartier instead of Weil divisors.
Another, more elementary way is to use Lemma 3.4 of [30]. It is shown in that lemma that there exists a family of polynomials P i,j (i = 0, . . . , l g − 1, j = 1, . . . , J) in the projective coordinates of s ∈ S and p ∈ A s ⊂ P l g −1 with the following properties: Every P i,j is a polynomial with integer coefficients, homogeneous of degree 2(l 8g − 1) in the coordinates of s and homogeneous of degree 4 in the coordinates of p. For every s ∈ S and p ∈ A s and every j ∈ {1, . . . , J}, the P i,j (s, p) (i = 0, . . . , l g − 1) are either all zero or they are the projective coordinates of 2p in A s ⊂ P l g −1 (by abuse of notation, P i,j (s, p) denotes P i,j evaluated at the projective coordinates of s and p). Furthermore, there exists j ∈ {1, . . . , J}, depending on s and p, such that not all P i,j (s, p) (i = 0, . . . , l g − 1) are zero.
Fixing j ∈ {1, . . . , J} and following the proof of Theorem B.2.5(a) in [22] (which amounts to the triangle inequality), we get a bound of the form where c 6 depends only on l, g and the (integral) coefficients of the P i,j , but is independent of s and p. The bound is valid for those s and p, where not all P i,j (s, p) (i = 0, . . . , l g − 1) are zero. After reiterating the process for every j ∈ {1, . . . , J} and adjusting the constants if necessary, we can assume that the inequality holds for all s ∈ S and p ∈ A s . We then obtain easily from h s (p) = lim n→∞ where we used that ∞ n=1 4 −n = 1 3 . Let now p be a point of C as in the lemma. In view of the above inequality, it suffices to show that h s (p) h S (s). Since C is irreducible and not contained in A s , the morphism π| C : C → S is quasi-finite. It is also proper, hence finite. Therefore, the pullback π * L S of the ample line bundle L S is also ample.
On the other hand, the closed immersion ι : C → A g,l yields a very ample line bundle ι * L on C. It follows from the ampleness of π * L S that there exists some natural number N ∈ N such that π * L ⊗N S ⊗ ι * L ⊗(−1) is ample. If we choose associated heights h C,ι * L and h C,π * L S , it now follows from fundamental properties of the Weil height that , which implies together with Lemma 4.1 and Lemma 4.
We note that q is defined over a field extension of K(p) of degree at most η(g) deg φ s for a certain function η : N → N, since φ s is defined over a field extension of K(s) ⊂ K(p) of degree at most η(g) by Rémond's Théorème 1.2 in [52] and q has degree at most deg φ s over the compositum of K(p) and the field of definition of φ s , since all its Galois conjugates over that field lie in φ −1 s (p) and this fiber has deg φ s elements. Here, Rémond has obtained the best possible η, while the fact that the bound depends only on g goes back to Silverberg in [55] and Masser-Wüstholz in [30], Lemma 2.1. Hence, we have [K(q) : K] [K(p) : K].
Let Ω be a finite set of abelian varieties overQ such that every quotient A r+1 0 /H for some abelian subvariety H of A r+1 0 is isogenous overQ to some element of Ω. For each A ∈ Ω we can fix some norm · A on Hom(A r+1 0 , A ) ⊗ R and a symmetric ample line bundle on A to obtain a canonical height h A on A . After passing to a finite field extension, we can assume that all A ∈ Ω, all these line bundles as well as all elements of Hom(A r+1 0 , A ) for all A ∈ Ω are defined over K. Going through the proof of Proposition 9.1 in [20], we see that B is obtained as the irreducible component of ker α containing the neutral element for a surjective homomorphism α : A r+1 0 → A for some A ∈ Ω. If we write · = · A , then we even obtain from Lemma 9.5 of [20] a surjective homomorphism α : A r+1 0 → A such that B is the irreducible component of ker α containing the neutral element and α [K(p) : K]. We have a projection morphism ψ : B → A r 0 given by omitting the first coordinate. We let B = ψ(B) ⊂ A r 0 and let B 2 be the connected component of ker ψ = B ∩ (A 0 × {0} r ) ⊂ B containing the neutral element. Since q ∈ Γ + B 0 , it follows that B 2 ⊂ B 0 × {0} r . By Poincaré's reducibility theorem, there exists an abelian subvariety B 3 ⊂ B such that the restriction of the natural addition morphism B 2 × B 3 → B is an isogeny. It follows that ψ| B 3 : B 3 → B must be an isogeny. As usual, there exists an isogeny Since ψ| B 3 : B 3 → B is surjective, we can choose u ∈ B 3 such that ψ(u) = µ(γ 1 , . . . , γ r ). Applying Poincaré's reducibility theorem again, we find an abelian subvariety B ⊂ A r 0 such that the restriction of the natural addition morphism B × B → A r 0 is an isogeny. Again, we get an isogeny ρ : A r 0 → B × B in the other direction such that their composition is multiplication by a scalar. By projecting to the first coordinate, we obtain ρ : A r 0 → B . Let w ∈ A r 0 with ρ(w) = (µ(γ 1 , . . . , γ r ), 0). It follows that µ(γ 1 , . . . , γ r ) is some multiple of w and hence w ∈ Γ r . We have (deg ψ| It follows from ψ(u) = µ(γ 1 , . . . , γ r ) that µ(q, γ 1 , . . . , γ r ) ∈ u + ker ψ ⊂ u + (A 0 ) r+1 tors + B 2 and by considering only the first coordinate we see that µq ∈ π 1 (u) + (A 0 ) tors + π 1 (B 2 ) ⊂ Γ + π 1 (B 2 ) and hence q ∈ Γ + π 1 (B 2 ). Now, B 1 = π 1 (B 2 ) is an abelian subvariety of A 0 of degree deg B 1 = deg B 2 with respect to L 0 . Since B 2 is an irreducible component of B ∩ (A 0 × {0} r ), we know that deg B 2 deg B by Proposition 3.1 of [49]. We also know that B 2 = B 1 × {0} r and so B 1 ⊂ B 0 , since B 2 ⊂ B 0 × {0} r . This proves (ii).
Since Hom(A r+1 0 , A) is a finitely generated Z-module and the height is quadratic, there exists a constant c 0 , depending only on the two abelian varieties and the choices of symmetric ample line bundles as well as the choice of the norm, such that h A (α (x)) ≤ c 0 α 2 r+1 i=1 h A 0 (x i ) for all α ∈ Hom(A r+1 0 , A) and all x = (x 1 , . . . , x r+1 ) ∈ A r+1 0 . In particular, this bound holds for our α as chosen above.
We apply Rémond's Lemme 6.1 in [51] to choose γ ∈ Γ and b ∈ B 1 such that Note that we have assumed Γ = Γ sat in Rémond's notation and that Rémond's Lemme also holds for = 0 as is the case here. Suppose that mγ = m 1 γ 1 + . . . + m r γ r with m ∈ N, m 1 , . . . , m r ∈ Z. Since h A 0 (γ ) [K(p) : K], h A 0 extends to a norm on Γ ⊗ R and all norms on the finite-dimensional R-vector space Γ ⊗ R are equivalent, we also have that max i=1,...,r For given N ∈ N, we can find n ≤ N and a 1 , . . . , a r ∈ Z such that max i=1,...,r a i − nm i m ≤ N 1 r −1 . It follows that where the constant c 9 depends only on A 0 , L 0 , A, the choice of symmetric ample line bundle on A as well as of the norm · on Hom(A r+1 0 , A) ⊗ R and

o-Minimality
We give a brief introduction to the theory of o-minimal structures and define all terms which are relevant in our application. We refer to the book of van den Dries ( [57]) for a more thorough treatment of o-minimal structures. 2. An o-minimal structure S (over (R, +, −, ·, <, 0, 1)) is a sequence S = (S n ) n∈N such that S n is a subset of the power set of R n for all n ∈ N and the following conditions are satisfied: . We call the elements of n∈N S n the definable sets with respect to S or simply the definable sets (if S is fixed).
Since our uniformization map goes to a product of projective spaces, we need to introduce the notion of a definable space. This notion is treated in more detail by van den Dries in Chapter 10 of [57]. In the following definitions, definability will always mean definability with respect to some fixed o-minimal structure S.
We call the f i charts of S. It is easily seen that image and pre-image of a definable set under a definable map or a morphism are definable and that the composition of two definable maps or morphisms is again a definable map or a morphism respectively. A definable map is a morphism with respect to the standard global charts of its domain and its range precisely if it is continuous.
By the Seidenberg-Tarski theorem, the semialgebraic sets themselves form an o-minimal structure (the definable maps of which are the semialgebraic maps). For our purposes, this will not be sufficient and we will have to work in the structure R an,exp , which contains (among other things) the graph of the exponential function on the real numbers and the graph of the restriction of any analytic function, defined on an open neighbourhood of [0, 1] n , to [0, 1] n (n ∈ N). That this structure is o-minimal and admits analytic cell decomposition is due to van den Dries and Miller (see [58]).
In order to prove our main theorem, we will need that rational points on definable sets are sparse unless there is a "reason" for them not to be sparse in the form of a semialgebraic set, contained in the definable set. This is the famous Pila-Wilkie Theorem. We will use a variant by Habegger and Pila, counting "semirational" points, which is what we will need in the proof.
(iii) If the o-minimal structure admits analytic cell decomposition, the restriction of δ to (0, 1) is real analytic.
A priori, the corollary only provides δ such that (π 2 , π 3 ) • δ is non-constant. Going through its proof, we see however that δ can actually be chosen such that π 3 •δ is non-constant. Note that we don't need the additional uniformity in families that the corollary provides.

Definability
In order to be able to use the powerful o-minimality result from the last section, we must show that our analytic uniformization of A g,l (C) is definable, when restricted to a suitable set. In order to be able to speak of e.g. definable or semialgebraic subsets of C or H g , we will always identify C with The following important proposition is due to Peterzil-Starchenko. Note that P l g −1 (C) is a definable space with respect to its standard atlas. Proposition 6.1. (Peterzil-Starchenko) The map exp : H g ×C g → A g,l (C) ⊂ P l g −1 (C) × P l g −1 (C), defined as in Proposition 2.1, has the following properties: (i) There is an open subset U of H g × C g such that the restriction of exp to U is a morphism of definable spaces in R an,exp and U contains the set where F is a Siegel fundamental domain for the congruence subgroup G(l, 2l) of Sp 2g (Z). (ii) The map exp | U is surjective.
Proof. Going back to the proof of Proposition 2.1, we see that it suffices to show that the function φ as defined there is definable, when restricted to an open set that contains This is a consequence of Corollary 7.10(1) of [39] with D = lE g , since F consists of finitely many translates of the Siegel fundamental domain and As exp is clearly continous, we deduce (i).
Next, we deduce (ii) from Proposition 2.1(ii), since U contains at least one element of each orbit of the action of G(l, 2l) Z 2g on H g × C g .

Functional transcendence
Let S ⊂ A g,l be an irreducible smooth locally closed curve, set A = π −1 (S) and let C ⊂ A be an irreducible closed curve. Let exp be as in Proposition 2.1. Once we have used the Habegger-Pila theorem to find a semialgebraic obstruction, the following theorem (known as "Ax of log type") which is due to Gao will allow us to conclude that C is contained in an irreducible variety as described in Theorem 1.2 of suitable codimension.
Recall that ξ is the generic point of S and AQ (S)/Q ξ , Tr is theQ(S)/Qtrace of A ξ . In this section, we will use subscripts to denote the base change of varieties and morphisms. One irreducible component of the variety W in Theorem 7.1 is the variety the existence of which Theorem 1.2 postulates. Our statement of the theorem differs from Gao's in the terminology that we use. Before we can prove that our version follows from Gao's version, we need to introduce Gao's terminology: We follow the exposition in [14]. An abelian subscheme B of A C is an irreducible subgroup scheme of A C → S C which is proper and flat over S C and dominates S C . An irreducible subvariety Z of A C is called a generically special subvariety of sg type if there exists a finite cover S → S C , inducing a morphism ρ : A = A C × S C S → A C such that Z = ρ(σ + σ 0 + B ), where B is an abelian subscheme of A , σ is a torsion section of A and σ 0 is a constant section of A , i.e. the composition of a section S → C × S , s → (q, s) (C an abelian variety over C, q ∈ C (C)) with an isomorphism between C × S and an abelian subscheme of A . We can now prove Theorem 7.1.
Proof. We apply Theorem 8.1 of [17] to the connected mixed Shimura variety S = A g,l (C) with uniformization map exp : H g × C g → A g,l (C) and subvariety Y equal to the Zariski closure of C(C) in A g,l (C). As Y (C)\C(C) is finite, the closure with respect to the Euclidean topology ofỸ is a complex analytic irreducible component of exp −1 (Y (C)). Together with Theorem 8.1 of [17], this implies that exactly one complex analytic irreducible component of the intersection of the Zariski closure ofỸ in M g (C) × C g with H g × C g containsỸ and that this component maps onto the complex points of a weakly special subvarietyW of (A g,l ) C and thatW is the smallest weakly special subvariety containing Y . As a weakly special subvariety,W is irreducible. By Proposition 5.3 of [14] (cf. Proposition 1.1 and 3.3 of [16]) the varietyW is a generically special subvariety of sg type of the abelian scheme π −1 (π(W)) → π(W), where this term is defined analogously for π −1 (π(W)) → π(W) as for A C → S C (see Definition 1.5 in [14]).
A priori,W is defined over C, but since it is the smallest such weakly special subvariety, Galois conjugates of weakly special subvarieties as well as irreducible components of intersections of weakly special subvarieties are weakly special and C and hence Y are defined overQ, it must be defined overQ. We set W =W ∩ A, considered as a variety overQ. This is a subvariety of A that contains C.
Let L be an algebraic closure of the function field of S C . We identifȳ Q(S) with the algebraic closure ofQ(S) in L. The L/C-trace of (A ξ ) L coincides with the base change of the CQ(S)/C-trace of (A ξ ) CQ(S) , which coincides with the base change of theQ(S)/Q-trace of A ξ by Theorem 6.8 of [10]. AsW is generically special of sg type (as a variety over C), it follows from the universal property of the trace that every irreducible component of (W ξ ) L is a translate of an abelian subvariety of (A ξ ) L by a point in Note that Proposition 5.3 of [14] applies only to the universal family of principally polarized abelian varieties with symplectic level l-structure, but the same statement can be proved analogously for any connected mixed Shimura variety of Kuga type coming from a neat congruence subgroup, so in particular for A g,l (C) (see Proposition 1.2.14 and Corollary 1.2.15 in Gao's dissertation [15]). One could also apply Proposition 5.3 of [14] to an irreducible component of the preimage ofW(C) under the canonical Shimura morphism from the universal family of principally polarized abelian varieties with symplectic level 2l-structure to A g,l (C). Lemma 8.1. We can assume without loss of generality that S ⊂ A g,l is a smooth irreducible locally closed curve (not necessarily closed in A g,l ) and A = π −1 (S).
Proof. For l big enough, the scheme A g,l with the family of abelian varieties A g,l → A g,l is the fine moduli scheme of principally polarized abelian varieties of dimension g with level structure "between l and 2l". For the precise moduli interpretation, see [36], Appendix to Chapter 7, Section B. In particular, if our family is a pull-back of the universal family of principally polarized abelian varieties of dimension g with symplectic level 2l-structure, it will automatically also be a pull-back of A g,l → A g,l .
Let therefore A → S for the moment be an arbitrary abelian scheme over an irreducible smooth curve of relative dimension g. If ξ is the generic point of S, then the abelian variety A ξ is isogenous to a principally polarized abelian varietyÃ. The abelian varietyÃ as well as the isogeny are defined over some finite extension F ofQ(S). After replacing S by a finite cover S → S and A by its pullback under that cover, we may assume that F =Q(S). We can replace S by a finite cover, since an irreducible subvariety W ⊂ A × S S as described in Theorem 1.2 projects to an irreducible subvariety of A of the same form. By Theorem 3 of Section 1.4 in [8], there exists a Néron modelÃ ofÃ over S as defined in Definition 1 of Section 1.2 in [8]. By the universal property of the Néron model, we obtain an S-morphism A →Ã which extends the isogeny between A ξ andÃ.
By Theorem 3 of Section 1.4 in [8], there is a Zariski open subsetS of S such thatÃ × SS is an abelian scheme overS. SinceS is smooth, it follows as in [13], p. 6, that the abelian schemeÃ × SS is principally polarized, i.e. admits an isomorphism of group schemes overS to its dual abelian scheme such that the restriction of the isomorphism to each fiber over a geometric point ofS is induced by an ample line bundle on that fiber. The morphism between A andÃ that extends the isogeny between A ξ andÃ is dominant and proper, hence surjective, so its restriction to each fiber over a point inS is an isogeny. We see that it suffices to prove the theorem forÃ × SS →S, hence we can assume that A is a principally polarized abelian scheme. We can then add symplectic level 2l-structure to the family A → S by taking a finite cover of S (corresponding to the finite field extension ofQ(S) that is obtained by adding the 2l-torsion points of the generic fiber).
Having done this, there is a cartesian diagram where the morphisms i and i S are defined overQ. This is a consequence of Theorem 7.9 in [36] that asserts the existence of a fine moduli space for principally polarized abelian varieties of dimension g with full level lstructure for l big enough (in fact, l ≥ 3 suffices). The family A is then a pullback of the universal family with symplectic level 2l-structure and therefore also of the family A g,l → A g,l (cf. [36], Appendix to Chapter 7, Section B). For every s ∈ S, the restriction i| As is an isomorphism between A s and i(A s ).
If the family A is not isotrivial, as we suppose in our theorem, the map i S is non-constant, so has finite fibers, and therefore i has finite fibers as well.
Thus, the curve i(C) must intersect the enlarged isogeny orbit in infinitely many points as well. If W ⊂ i(A) is of the form described in Theorem 1.2, then every irreducible component of i −1 (W) ⊂ A that dominates S is as well, so it suffices to prove our theorem for i(A) → i S (S). We can even pass to a Zariski open smooth subset of i S (S) (we use that i(C) intersects every fiber in only finitely many points). This proves the lemma. 8.1.2. Producing many points of bounded height. We now return to subfamilies of A g,l → A g,l of the form π −1 (S) → S with S smooth, irreducible and locally closed. We will keep the same notation until the end of the proof.
We have sup Γ ∩ C would be finite as well, since C intersects every fiber of π in only finitely many points.
For each s ∈ S such that A 0 and A s are isogenous, let φ s : A 0 → A s be the isogeny furnished by Corollary 3.4. We choose a point p ∈ A [k] Γ ∩ C. Thanks to Lemma 2.2, we can write p = φ π(p) (γ + b) for some γ ∈ Γ, b ∈ B 0 with B 0 an abelian subvariety of A 0 of codimension ≥ k. We set s = π(p), d = [K(p) : K]. By the above, we can make d arbitrarily big with the right choice of p. If σ is an element of Gal(Q/K), then it follows that σ(p) = σ(φ s )(σ(γ) + σ(b)), where σ acts on algebraic points and maps in the usual way.
As C and S are defined over K, the points σ(p) and σ(s) lie again on C and S respectively. Note that the addition morphism A 0 × A 0 → A 0 and the inversion morphism A 0 → A 0 are both defined over K -in particular, σ fixes the zero element of A 0 . Furthermore, it sends the zero element of A s to the zero element of A σ(s) . It follows that the map σ(φ s ) is an isogeny between σ(A 0 ) = A 0 and σ(A s ) = A σ(s) with kernel σ(ker φ s ) and therefore has degree deg σ(φ s ) = deg φ s . Since we have assumed that all endomorphisms of A 0 are defined over K, we have σ(B 0 ) = B 0 .
Thus, we get d different points σ(p) in A [k] Γ ∩ C. Each of these points has some pre-image (τ σ , p σ ) in U under exp | U because of Proposition 6.1(ii), where exp and U are defined as in that same proposition. From the proof of Proposition 6.1(ii), we see that we can choose τ σ in a Siegel fundamental domain for G(l, 2l) and p σ in a corresponding fundamental parallelogram for the lattice τ σ Z g + Z g , i.e. p σ = Ω τσ x σ with x σ ∈ [0, 1) 2g .
where the implicit constants are independent of p, σ and d.

8.1.3.
Application of the point-counting theorem. From now on, "definable" will always mean "definable in the o-minimal structure R an,exp ". Let exp and U be defined as in Proposition 6.1. The set X = exp | −1 U (C(C)) ⊂ H g × C g is definable as C(C) is semialgebraic, being a quasiprojective algebraic curve, and exp | U is definable by Proposition 6.1(i). Lemma 8.3. There exists a non-constant real analytic map α : (0, 1) → X such that the transcendence degree over C of the field generated by its complex coordinate functions is at most g − k + 1.
It follows from and AΩ τ 0 = Ω τ B that first τ •δ and then A•δ are semialgebraic as well. Here and in the following, we use variables like τ and A also for the corresponding coordinate functions on Z. We can then use that Ω τ 0 R + A 1γ1 + . . . + A rγr + Ω τ 0 Hy to deduce that Now H • δ is semialgebraic. For each t ∈ (0, 1) the rank of Ω τ 0 (H • δ)(t) is at most g−k by the definition of Z. Note that the first summand in the above expression for Ω τ x is semialgebraic when composed with δ. Therefore we can conclude that the transcendence degree over C of the complex coordinate functions of the real analytic map α = ψ • π 3 • δ : (0, 1) → X is at most g − k + 1, where ψ(τ, x) = (τ, Ω τ x). Since π 3 • δ is non-constant, so is α.
Since α is non-constant, we can choose some t ∈ (0, 1), where the derivative of α doesn't vanish. Since the Taylor series of α in t must have positive radius of convergence, we can find some holomorphic mapα : D → X from a small open disk D to X such that t ∈ D andα| D∩(0,1) = α| D∩(0,1) . By the identity theorem for holomorphic functions, it follows that the transcendence degree over C of the coordinate functions ofα is at most g − k + 1 as well.
As the derivative of α doesn't vanish at t, the mapα is non-constant as well. Since every complex analytic irreducible component of X has complex dimension 1, it follows by analytic continuation of the algebraic relations between the coordinate functions ofα along the corresponding complex analytic irreducible component of exp −1 (C(C)) that the Zariski closure of this complex analytic irreducible component of exp −1 (C(C)) inside M g (C) × C g has dimension at most g − k + 1. Theorem 1.2 now follows from Theorem 7.1.

8.2.
Proof of Theorem 1.3. If π(C) = S, we can apply Theorem 1.2 with k = g. If π(C) = S, there exists s ∈ S such that C ⊂ A s and now we can apply the non-relative Mordell-Lang conjecture which Raynaud proved in this case in [48] by reducing it to the theorem of Faltings to conclude that A Γ ∩ C = φ s (Γ) ∩ C is finite unless C is equal to a translate of an abelian subvariety by a point of φ s (Γ) ⊂ A Γ . This argument works for any family A → S, not only for subfamilies of A g,l → A g,l : We need only Lemma 2.2 in order to fix the isogeny and this also holds for any family after maybe enlarging Γ. If C is then a translate of an abelian subvariety of A s , it is a translate of that abelian subvariety by any point on C and hence also by a point in the isogeny orbit of the original Γ. 8.3. Proof of Corollary 1.4. Suppose that C ∩ (Σ × Γ ) is infinite. Let S be the smooth locus of pr 1 (C) ⊂ A g,l . We can assume without loss of generality that dim S = 1, otherwise pr 1 is constant and we are done. Let π : A g,l → A g,l be the natural morphism as in Section 2 and let : A g,l → A g,l be the zero section.
If ξ denotes the generic point of S, then we have A ⊂ AQ (S)/Q ξ . Since C dominates S, it doesn't satisfy condition (i) of Theorem 1.3, so it has to satisfy condition (ii). This implies that the projection of C to S × A must be the graph of a constant map S → A. We deduce that pr 2 is constant.