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The Betti map associated to a section of an abelian scheme

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Abstract

Given a point \(\xi \) on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of \(\xi \). When \((A, \xi )\) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often \(\xi \) takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when \(\xi \) is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to \((A, \xi )\) (assuming A without fixed part, and \({\mathbb {Z}}\xi \) Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension \(\le 3\), and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if \(A\rightarrow S\) is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space \({{\mathcal {A}}}_g\) has dimension at least g, then the Betti map of any non-torsion section \(\xi \) is generically a submersion, so that \(\,\xi ^{-1}A_{\mathrm{tors}}\) is dense in \(S({\mathbb {C}})\).

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Notes

  1. Formulated in [29] and further discussed in [30]; see also the third author’s book [37], especially chapter 3).

  2. For the convoluted story of the proof of this theorem, see for instance [3, §1], [5].

  3. Actually, it also depends on a choice of branch of abelian logarithm \(\lambda \), cf. Sect. 3.1.

  4. We do not know whether the converse holds: the image of \(\beta \) might be dense without containing a dense open subset.

  5. As Z. Gao pointed out [13], this guess should be slightly amended to take into account the fact that the right hand side of (2.1) is not always additive in A/S.

  6. This is the criterion used in [35, th. 0.6]: the proof ot this criterion is presented in [35, §3] in a language more congenial to Hodge theoretists.

  7. In a previous version, an explicit link of these issues with the “Ax-Schanuel conjecture” was pointed out. Very recently this conjecture has been proved, making thus possible to apply it to our problem, as done by Gao in the appendix.

  8. This terminology, due to D. Bertrand, refers to the fact that these are indeed real-analytic coordinates on any simply-connected domain in \(A({\mathbb {C}})\), and that \({\mathcal {L}}_\xi \) is nothing but the Betti realization of the 1-motive \([{\mathbb {Z}}{\mathop {\rightarrow }\limits ^{1\mapsto \xi }} A]\) attached to \((A, \xi )\) in the sense of [9, X].

  9. As in [4], we write this differential system in a slightly non-standard way, with the matrix of the connection on the right so that the monodromy acts on the left. \(\ell _\partial \) is essentially the row of rational functions which occur as second member of the inhomogeneous scalar differential operator in Manin’s kernel theorem.

  10. In this respect, the presentation of [2, Th. 3] in [35, §3] is an overinterpretation.

  11. We are grateful to M. Brion for this reference.

  12. More convenient here than the usual convention 1, 2, 3, 4: but \(A_1 = C_1\) and \(B_2= C_2\).

  13. Gao has shown [13] that this construction leads to a counter-example to the conclusion of Theorem 2.3.1 if we drop the assumptions.

  14. By Hodge theory, \({\mathbb {G}}_m = Z({GSp}_{2g})\) is contained in \(\mathrm {MT}(S)\). In this paper we only need \({Sp}_{2g} < \mathrm {MT}(S)\).

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Acknowledgements

We are pleased to thank Professor C. Voisin for sharing with us preliminary versions of her paper [35] and for interesting conversations with all of us. We thank Professors D. Bertrand, Ph. Griffiths, I. Krichever and B. Mazur for their interest in this work and helpful interactions. The third author thanks Professor J-P. Serre for several helpful comments, in particular pointing out inaccuracies in previous versions of Sect. 9 and suggesting Theorem 9.1.1 in its present form. This research was partially supported by the grant DIMAGeometry PRIDZUCC of the University of Udine.

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Appendix by Z. Gao: an application of the pure Ax-Schanuel Theorem

Institut Mathématique de Jussieu - Paris Rive Gauche, 4, Place Jussieu, 75005, Paris, France. E-mail address: ziyang.gao@imj-prg.fr.

Appendix by Z. Gao: an application of the pure Ax-Schanuel Theorem

1.1 Main result

Let \({{\mathcal {A}}}_g\) be the moduli space of principally polarized abelian varieties of dimension g, possibly with some level structure.

Let S be a complex irreducible algebraic variety. Let \(\pi {:}\; A\rightarrow S\) be an abelian scheme of relative dimension g. We may assume that A/S is principally polarized up to replacing A by an isogeneous abelian scheme. Then \(\pi {:}\; A\rightarrow S\) induces a modular map \(\mu _A {:}\; S\rightarrow {{\mathcal {A}}}_g\). We assume \(\dim \mu _A(S) \ge g\).

Next we want to understand when S satisfies the conclusion of Corollary 2.2.2. More precisely, let \(\mathfrak {H}_g\) be the Siegel upper half space and let \({u} : \mathfrak {H}_g \rightarrow {{\mathcal {A}}}_g\) be the uniformization. Denote by \(\tilde{S}\) a complex analytic irreducible component of \({u}^{-1}(\mu _A(S))\) in \(\mathfrak {H}_g\). We name

For any \(\tilde{s} \in \tilde{S}\)and any \({\mathbf {c}} \in {\mathbb {C}}^g\), there exists a complex analytic subvariety \(\tilde{C} \subset \tilde{U}\)of dimension \(\dim \mu _A(S) - g + 1\)passing through \(\tilde{s} \in \mathfrak {H}_g\)such that \(\tilde{s}' {\mathbf {c}}\)is constant for any \(\tilde{s}' \in \tilde{C}\). Here we view \({\mathbf {c}} \in {\mathbb {C}}^g\)as a column vector and \(\tilde{s}' {\mathbf {c}}\)is the usual matrix product (recall that every point in \(\mathfrak {H}_g\)is a \(g\times g\)matrix).

Our main result is:

10.1.1 Theorem

If Condition ACZ is satisfied, then \(\mu _A(S)\) is contained in a proper special subvariety of \({{\mathcal {A}}}_g\).

10.1.2 Remark

There are several equivalent ways to state the conclusion of Theorem 10.1.1. In fact for any irreducible subvariety S of \({{\mathcal {A}}}_g\), we shall prove in Lemma 10.2.6 that the following statements are equivalent:

  1. (i)

    The variety S is not contained in any proper special subvariety of \({{\mathcal {A}}}_g\).

  2. (ii)

    The variety S contains a point with Mumford-Tate group \({GSp}_{2g}\).

  3. (iii)

    The monodromy group of A/S is Zariski dense in \({Sp}_{2g}\).

In practice, condition (i) is often checked by computation of the Mumford-Tate group (hence condition (ii)) or of the monodromy group (hence condition (iii)). For example as A/S is not isotrivial by Assumption 1, if we denote by \(V = H^1(A_s,{\mathbb {C}})\) for any \(s \in S\), then condition (iii) holds if (and only if) the symmetric square \(S^2 V\) is irreducible for the monodromy action by Beukers-Brownawell-Heckman [7, Theorem 2.2].

In fact, Theorem 10.1.1 can be deduced from a more technical and more general theorem. In order to state the theorem we need to introduce some notation.

Recall the uniformization \({u} : \mathfrak {H}_g \rightarrow {{\mathcal {A}}}_g\). The natural embedding \(\mathfrak {H}_g \subset {\mathbb {C}}^{g(g+1)/2}\) endows \(\mathfrak {H}_g\) with a structure of “complex algebraic variety”, and hence u gives rise to a bi-algebraic system. Say a complex analytic irreducible subset \(\tilde{Y}\) of \(\mathfrak {H}_g\) is bi-algebraic if \(\tilde{Y}\) is algebraic in \(\mathfrak {H}_g\) and \({u}(\tilde{Y})\) is algebraic in \({{\mathcal {A}}}_g\). See Sect. 10.2 for more details.

Denote by \(H_S^{\circ }\) the connected algebraic monodromy group of A/S, namely \(H_S^{\circ }\) is the neutral component of the Zariski closure of \(\mathrm {im}(\pi _1(S,s) \rightarrow \pi _1({{\mathcal {A}}}_g,s)) \subset {Sp}_{2g}({\mathbb {Z}})\) in \({Sp}_{2g}\). Denote by \(\tilde{S}^{\mathrm {biZar}}\) the smallest bi-algebraic subset of \(\mathfrak {H}_g\) which contains \(\tilde{S}\). It exists by Lemma 10.2.3.

Recall that every element of \(\mathfrak {H}_g\) is a \(g\times g\)-matrix. For any \({\mathbf {c}} \in {\mathbb {C}}^g\) and any \(\tilde{s} \in \mathfrak {H}_g\), denote by

$$\begin{aligned} H_{{\mathbf {c}},\tilde{s}} := \{Z \in \mathfrak {H}_g: Z {\mathbf {c}} = \tilde{s} {\mathbf {c}}\}. \end{aligned}$$

10.1.3 Theorem

There does not exist an abelian scheme A/S with \(\dim \mu _A(S) \ge g\) satisfying the following three properties:

  1. (i)

    The connected algebraic monodromy group \(H_S^{\circ }\) is simple;

  2. (ii)

    There exist \({\mathbf {c}} \in {\mathbb {C}}^g\) and \(\tilde{s} \in \tilde{S}\) such that \(\mathrm {codim}_{\tilde{S}^{\mathrm {biZar}}}(H_{{\mathbf {c}},\tilde{s}} \cap \tilde{S}^{\mathrm {biZar}}) = g\).

  3. (iii)

    Condition ACZ is satisfied.

1.2 Review on the bi-algebraic system of \({{\mathcal {A}}}_g\) and Ax-Schanuel

We focus on the case \({{\mathcal {A}}}_g\). We shall consider the uniformization \({u} {:}\; \mathfrak {H}_g \rightarrow {{\mathcal {A}}}_g\), where \(\mathfrak {H}_g\) is the Siegel upper half space defined as the following.

$$\begin{aligned} \mathfrak {H}_g = \left\{ Z = X + \sqrt{-1}Y \in \mathrm {Mat}_{g\times g}({\mathbb {C}}) : Z = Z^{\mathrm {t}},~ Y>0 \right\} . \end{aligned}$$

Later on we will study subvarieties of \(\mathfrak {H}_g\) and \({{\mathcal {A}}}_g\) at the same time. To distinguish them we often use letters to denote subsets of \({{\mathcal {A}}}_g\) and add a \(\sim \) on top to denote subsets of \(\mathfrak {H}_g\).

Consider

$$\begin{aligned} {\mathfrak {p}}_g = \left\{ Z \in \mathrm {Mat}_{g\times g}({\mathbb {C}}) : Z = Z^{\mathrm {t}} \right\} . \end{aligned}$$

Then \({\mathfrak {p}}_g \cong {\mathbb {C}}^{g(g+1)/2}\) as \({\mathbb {C}}\)-vector spaces. The Siegel upper half space is open (in the usual topology) and semi-algebraic in \({\mathfrak {p}}_g\). The complex structure on \({\mathfrak {p}}_g\) thus induces a structure of complex analytic variety on \(\mathfrak {H}_g\). Following Pila, Ullmo and Yafaev, we define

10.2.1 Definition

A subset \(\tilde{Y}\) of \(\mathfrak {H}_g\) is said to be irreducible algebraic if it is a complex analytic irreducible component of \(\mathfrak {H}_g \cap \tilde{Y}^{\mathrm {c}}\) for some algebraic subvariety \(\tilde{Y}^{\mathrm {c}}\) of \({\mathfrak {p}}_g\).

Hence we have the following definition.

10.2.2 Definition

  1. (1)

    A subset \(\tilde{F}\) of \(\mathfrak {H}_g\) is said to be bi-algebraic if it is irreducible algebraic in \(\mathfrak {H}_g\) and \({u}(\tilde{F})\) is an algebraic subvariety of \({{\mathcal {A}}}_g\).

  2. (2)

    An irreducible subvariety F of \({{\mathcal {A}}}_g\) is said to be bi-algebraic if one (and hence any) complex analytic irreducible component of \({u}^{-1}(F)\) is irreducible algebraic in \(\mathfrak {H}_g\).

Before moving on, let us make the following observation.

10.2.3 Lemma

Let \(F_1\) and \(F_2\) be two bi-algebraic subvarieties of \({{\mathcal {A}}}_g\), and let F be an irreducible component of \(F_1 \cap F_2\). Then F is also bi-algebraic.

Proof

First F is clearly irreducible algebraic. Consider a complex analytic irreducible component \(\tilde{F}\) of \({u}^{-1}(F)\). It is contained in both \(\tilde{F}_1\), an irreducible component of \({u}^{-1}(F_1)\), and \(\tilde{F}_2\), an irreducible component of \({u}^{-1}(F_2)\). Let \(\tilde{F}'\) be an irreducible component of \(\tilde{F}_1 \cap \tilde{F}_2\) which contains \(\tilde{F}\). Then \( F = {u}(\tilde{F}) = {u}(\tilde{F}') \subset {u}(\tilde{F}_1) \cap {u}(\tilde{F}_2) = F_1 \cap F_2. \) Taking the Zariski closures, we get \( F \subset {u}(\tilde{F}')^{{Zar}} \subset F_1 \cap F_2. \) Now since \(\tilde{F}'\) is irreducible, we know that \({u}(\tilde{F}')^{{Zar}}\) is irreducible. Hence \(F = {u}(\tilde{F}')^{{Zar}}\) since F is an irreducible component of \(F_1 \cap F_2\). Therefore \(F = {u}(\tilde{F}')\) and so \(\tilde{F} = \tilde{F}'\) is algebraic. So F is bi-algebraic. \(\square \)

Based on this observation, for any irreducible subvariety Y of \({{\mathcal {A}}}_g\), there exists a unique smallest bi-algebraic subvariety of \({{\mathcal {A}}}_g\) which contains Y. Then for any complex analytic irreducible subset \(\tilde{Y}\) of \(\mathfrak {H}_g\), there exists a unique smallest bi-algebraic subset of \(\mathfrak {H}_g\) which contains \(\tilde{Y}\): it is an irreducible component of the smallest bi-algebraic subvariety of \({{\mathcal {A}}}_g\) which contains \({u}(\tilde{Y})^{{Zar}}\).

There is a better characterization of bi-algebraic subvarieties of \({{\mathcal {A}}}_g\) using Hodge theory and group theory. They are precisely the so-called weakly special subvarieties of \({{\mathcal {A}}}_g\) defined by Pink [29, Definition 4.1.(b)]. This is proven by Ullmo–Yafaev [34, Theorem 1.2]. Moonen has also studied these subvarieties and proved that they are precisely the totally geodesic subvarieties of \({{\mathcal {A}}}_g\). See [25, 4.3]. Linearity properties in Shimura varieties was first studied by Moonen in loc.cit. For our purpose we prove the following lemma.

10.2.4 Lemma

Let \(\tilde{F}\) be a bi-algebraic subset of \(\mathfrak {H}_g\). Then it is affine linear, meaning that it is the intersection of \(\mathfrak {H}_g\) with some affine linear subspace of \({\mathfrak {p}}_g\).

Proof

This follows from Ullmo–Yafaev’s characterization and the Harish–Chandra realization of Hermitian symmetric domains. Let us explain the details. We use the language of Shimura data in the proof.

By a result of Ullmo–Yafaev [34, Theorem 1.2], bi-algebraic subsets of \(\mathfrak {H}_g\) are precisely the weakly special subsets of \(\mathfrak {H}_g\). Hence \(\tilde{F}\) is a weakly special subset of \(\mathfrak {H}_g\). By definition of weakly special subvarieties (see [34, Definition 2.1] or [29, Definition 4.1.(b)]), there exist a connected Shimura subdatum \((G,{{\mathcal {X}}})\) of \(({GSp}_{2g},\mathfrak {H}_g)\) and a decomposition \((G^{{ad}},{{\mathcal {X}}}) = (G_1,{{\mathcal {X}}}_1) \times (G_2,{{\mathcal {X}}}_2)\) and a point \(\tilde{x}_2 \in {{\mathcal {X}}}_2\) such that \(\tilde{F} = {{\mathcal {X}}}_1 \times \{\tilde{x}_2\}\).

Take any point \((\tilde{x}_1,\tilde{x}_2) \in {{\mathcal {X}}}_1 \times \{\tilde{x}_2\} \subset {{\mathcal {X}}}\), we have the Harish–Chandra embedding of \({{\mathcal {X}}}\) into \(T_{(\tilde{x}_1,\tilde{x}_2)} {{\mathcal {X}}}\), the tangent space of \({{\mathcal {X}}}\) at \((\tilde{x}_1,\tilde{x}_2)\). We have also the Harish–Chandra embedding of \(\mathfrak {H}_g\) into \(T_{(\tilde{x}_1,\tilde{x}_2)} \mathfrak {H}_g\). These two embeddings are compatible in the following sense: \(T_{(\tilde{x}_1,\tilde{x}_2)} {{\mathcal {X}}}\) is a linear subspace of \(T_{(\tilde{x}_1,\tilde{x}_2)} \mathfrak {H}_g\) and \({{\mathcal {X}}}= \mathfrak {H}_g \cap T_{(\tilde{x}_1,\tilde{x}_2)} {{\mathcal {X}}}\). This is proven in Helgason [16, Chapter VIII, \(\mathsection \)7]. We refer to [23, Chapter 5, \(\mathsection \)2, Theorem 1] for the presentation. In particular, the Harish–Chandra embedding realizes \(\mathfrak {H}_g\) as the unit ball in \({\mathbb {C}}^{g(g+1)/2}\).

The natural embedding of the Hermitian symmetric space \(\mathfrak {H}_g\) into \({\mathfrak {p}}_g\) can be realized as the Harish–Chandra embedding mentioned above composed with a linear transformation which we call \(\ell \). Define \({\mathfrak {p}} = \ell (T_{(\tilde{x}_1,\tilde{x}_2)} {{\mathcal {X}}})\). Then \({\mathfrak {p}}\) is an affine subspace of \({\mathfrak {p}}_g\). Now by the compatibility mentioned in the last paragraph, we have that \({{\mathcal {X}}}= {\mathfrak {p}} \cap \mathfrak {H}_g\).

The decomposition of Hermitian symmetric spaces \({{\mathcal {X}}}= {{\mathcal {X}}}_1 \times {{\mathcal {X}}}_2\) gives a decomposition \({\mathfrak {p}} = {\mathfrak {p}}_1 \times {\mathfrak {p}}_2\) as complex spaces, and \(\tilde{F} = {{\mathcal {X}}}_1 \times \{\tilde{x}_2\}\) is then \(({\mathfrak {p}}_1 \times \{0\}) \cap {{\mathcal {X}}}= ({\mathfrak {p}}_1 \times \{0\}) \cap \mathfrak {H}_g\). Hence we are done. \(\square \)

Now we are ready to state the Ax-Schanuel theorem for \({{\mathcal {A}}}_g\). It is recently proven by Mok–Pila–Tsimerman [24]. This theorem has several equivalent forms, whose equivalences are not hard to show. For our purpose we only need the following weak form.

10.2.5 Theorem

Let \(\tilde{Y}\) be an irreducible complex analytic subvariety of \(\mathfrak {H}_g\). Let \({\tilde{Y}}^{biZar}\) be the smallest bi-algebraic subset of \(\mathfrak {H}_g\) which contains \(\tilde{Y}\). Then

$$\begin{aligned} \dim \tilde{Y}^{{Zar}} + \dim {u}(\tilde{Y})^{{Zar}} \ge \dim \tilde{Y} + \dim {\tilde{Y}}^{biZar}. \end{aligned}$$

Here \(\tilde{Y}^{{Zar}}\) means the smallest irreducible algebraic subset of \(\mathfrak {H}_g\) which contains \(\tilde{Y}\).

We end this section by proving the equivalence of the following statements.

10.2.6 Lemma

Let S be an irreducible subvariety of \({{\mathcal {A}}}_g\). Then the following statements are equivalent:

  1. (i)

    The variety S is not contained in any proper special subvariety of \({{\mathcal {A}}}_g\).

  2. (ii)

    The variety S contains a point with Mumford-Tate group \({GSp}_{2g}\).

  3. (iii)

    The monodromy group of A/S is Zariski dense in \({Sp}_{2g}\).

  4. (iv)

    The variety S is not contained in any proper bi-algebraic subvariety of \({{\mathcal {A}}}_g\) of positive dimension.

  5. (v)

    There exists a point \(s \in S({\mathbb {C}})\) such that the following condition holds: s is not contained in any proper bi-algebraic subvariety of \({{\mathcal {A}}}_g\) of positive dimension.

Proof

By Deligne–André, a very general point in \(S({\mathbb {C}})\) has the same Mumford–Tate group which we denote by \(\mathrm {MT}(S)\). The Mumford–Tate group \(\mathrm {MT}(S)\) is a reductive group, and a subgroup of finite index of the monodromy group is contained in \(\mathrm {MT}(S)^{\mathrm {der}}\). Here “very general” means that the point is taken outside an at most countable union of proper subvarieties of S. We refer to [2, Lemma 4] for these facts.

Now let us prove (iii) \(\Rightarrow \) (i). If the monodromy group of A/S is Zariski dense in \({Sp}_{2g}\), then \({Sp}_{2g}\) is a subgroup of \(\mathrm {MT}(S)^{\mathrm {der}} < {GSp}_{2g}^{\mathrm {der}} = {Sp}_{2g}\). Hence \(\mathrm {MT}(S)^{\mathrm {der}} = {Sp}_{2g}\). So \(\mathrm {MT}(S) = {GSp}_{2g}\).Footnote 15

For (ii) \(\Rightarrow \) (iii), we use a stronger result of André. Let \(H_S^{\circ }\) be the neutral component of the Zariski closure of the monodromy group of A/S in \({GSp}_{2g}\). Then by [2, Theorem 1], we have that \(H_S^{\circ }\) is a non-trivial normal subgroup of \({GSp}_{2g}^{\mathrm {der}} = {Sp}_{2g}\). But \({Sp}_{2g}\) is simple, so \(H_S^{\circ } = {Sp}_{2g}\).

Let us prove (i) \(\Rightarrow \) (ii). The smallest special subvariety of \({{\mathcal {A}}}_g\) which contains S is defined by a Shimura subdatum with underlying group \(\mathrm {MT}(S)\). If \(\mathrm {MT}(S) \not = {GSp}_{2g}\), then the smallest special subvariety of \({{\mathcal {A}}}_g\) is not \({{\mathcal {A}}}_g\), which contradicts the assumption of (i).

We have (iv) \(\Rightarrow \) (i) since every special subvariety of \({{\mathcal {A}}}_g\) is bi-algebraic.

The implication (v) \(\Rightarrow \) (iv) is easy.

It remains to prove (ii) \(\Rightarrow \) (v). Let \(s \in S({\mathbb {C}})\) be such that \(\mathrm {MT}(s) = {GSp}_{2g}\). Take \(\tilde{s} \in {u}^{-1}(s)\). Let F be a bi-algebraic subvariety of \({{\mathcal {A}}}_g\) which contains s with \(\dim F > 0\). Let \(\tilde{F}\) be an irreducible component of \({u}^{-1}(F)\) which contains \(\tilde{s}\). It suffices to prove \(\tilde{F} = \mathfrak {H}_g\).

The proof goes as follows. By a result of Ullmo–Yafaev [34, Theorem 1.2], bi-algebraic subsets of \(\mathfrak {H}_g\) are precisely the weakly special subsets of \(\mathfrak {H}_g\). Hence \(\tilde{F}\) is a weakly special subset of \(\mathfrak {H}_g\). By definition of weakly special subvarieties (see [34, Definition 2.1] or [29, Definition 4.1.(b)]), there exist a connected Shimura subdatum \((G,{{\mathcal {X}}})\) of \(({GSp}_{2g},\mathfrak {H}_g)\) and a decomposition \((G^{\mathrm {ad}},{{\mathcal {X}}}) = (G_1,{{\mathcal {X}}}_1) \times (G_2,{{\mathcal {X}}}_2)\) and a point \(\tilde{x}_2 \in {{\mathcal {X}}}_2\) such that \(\tilde{F} = {{\mathcal {X}}}_1 \times \{\tilde{x}_2\}\). The condition \(\mathrm {MT}(\tilde{s}) = {GSp}_{2g}\) implies that the smallest Shimura subdatum of \(({GSp}_{2g},\mathfrak {H}_g)\) whose underlying space contains \(\tilde{s}\) is \(({GSp}_{2g},\mathfrak {H}_g)\). Therefore \((G,{{\mathcal {X}}}) = ({GSp}_{2g},\mathfrak {H}_g)\). But then \(G^{\mathrm {ad}} = {GSp}_{2g}^{\mathrm {ad}}\) is a simple group, and hence either \(\tilde{F} = \mathfrak {H}_g\) or \(\tilde{F}\) is a point. But \(\dim \tilde{F} > 0\), so \(\tilde{F}= \mathfrak {H}_g\). \(\square \)

1.3 Proof of Theorem 10.1.1

We may replace S by \(\mu _A(S)\) and hence assume that S is an irreducible subvariety of \({{\mathcal {A}}}_g\) of dimension \(\ge g\). Recall the uniformization \({u} {:}\; \mathfrak {H}_g \rightarrow {{\mathcal {A}}}_g\) and our convention that \(\tilde{S}\) is a complex analytic irreducible component of \(u^{-1}(S)\).

The key to prove Theorem 10.1.1 is the following proposition, whose proof uses Ax-Schanuel.

10.3.1 Proposition

Suppose Condition ACZ is satisfied. Then for any \(\tilde{s} \in \tilde{S}\), there exists a bi-algebraic subset \(\tilde{F}\) of positive dimension, properly contained in \(\mathfrak {H}_g\), such that \(\tilde{s} \in \tilde{F}\).

Proof

Fix a \({\mathbf {c}} \in {\mathbb {C}}^g\) and define the following subspace of \(\mathfrak {H}_g\)

$$\begin{aligned} H_{{\mathbf {c}},\tilde{s}} := \{Z \in \mathfrak {H}_g: Z {\mathbf {c}} = \tilde{s} {\mathbf {c}}\}. \end{aligned}$$

Then \(H_{{\mathbf {c}},\tilde{s}}\) has codimension g in \(\mathfrak {H}_g\). Apply Condition ACZ to this \(\tilde{s} \in \tilde{S}\) and \({\mathbf {c}} \in {\mathbb {C}}^g\). Hence we obtain a complex analytic variety \(\tilde{C}\) of dimension \(\dim S - g + 1\) passing through \(\tilde{s}\) such that \(\tilde{C} \subset \tilde{S} \cap H_{{\mathbf {c}},\tilde{s}}\). Now \(\tilde{C} \subset H_{{\mathbf {c}},\tilde{s}}\), so we have

$$\begin{aligned} \dim \tilde{C}^{{Zar}} \le \dim H_{{\mathbf {c}},\tilde{s}} = \dim \mathfrak {H}_g -g. \end{aligned}$$
(10.1)

On the other hand \(\tilde{C} \subset \tilde{S}\), so \({u}(\tilde{C}) \subset {u}(\tilde{S}) = S\). Hence

$$\begin{aligned} \dim {u}(\tilde{C})^{{Zar}} \le \dim S. \end{aligned}$$
(10.2)

Apply Ax-Schanuel, namely Theorem 10.2.5, to \(\tilde{C}\). We obtain

$$\begin{aligned} \dim \tilde{C}^{\mathrm {Zar}} + \dim {u}(\tilde{C})^{{Zar}} \ge \dim \tilde{C} + \dim \tilde{C}^{biZar}. \end{aligned}$$
(10.3)

Assume \(\tilde{C}^{biZar} = \mathfrak {H}_g\). Then we have \(\, (\dim \mathfrak {H}_g - g) + \dim S \ge \dim \tilde{C} + \dim \mathfrak {H}_g \,\) by (10.1), (10.2) and (10.3). But this cannot hold since \(\dim \tilde{C} = \dim S - g + 1 > 0\). Hence \(\tilde{C}^{biZar} \not = \mathfrak {H}_g\). On the other hand \(\dim \tilde{C}^{biZar} > 0\) since \(\dim \tilde{C} = \dim S - g + 1 \ge 1\). So we can take the desired \(\tilde{F}\) to be \(\tilde{C}^{biZar}\). \(\square \)

Before moving on, we point out that we have not yet used the full strength of Condition ACZ since we did not vary the variable \({\mathbf {c}}\). Now let us proof Theorem 10.1.1.

Proof of Theorem 10.1.1

Suppose Theorem 10.1.1 is not true. By condition (v) of Lemma 10.2.6, there exists a point \(s \in S({\mathbb {C}})\) such that s is not contained in any proper bi-algebraic subvariety of \({{\mathcal {A}}}_g\) of positive dimension. Take \(\tilde{s}\) to be a point in \({u}^{-1}(s)\) for this s. Applying Proposition 10.3.1 to \(\tilde{s}\), we get a bi-algebraic subset \(\tilde{F}\) of positive dimension, properly contained in \(\mathfrak {H}_g\), such that \(\tilde{s} \in \tilde{F}\). But then \({u}(\tilde{F})\) is a proper bi-algebraic subvariety of \({{\mathcal {A}}}_g\) of positive dimension which contains s. Now we get a contradition. \(\square \)

1.4 Proof of Theorem 10.1.3

In fact the same techniques for Theorem 10.1.1 can be used to prove Theorem 10.1.3.

Proof of Theorem 10.1.3

Suppose we have an abelian scheme A/S satisfying the three properties. Let \({\mathbf {c}} \in {\mathbb {C}}^g\) and \(\tilde{s} \in \tilde{S}\) be as in condition (ii) of Theorem 10.1.3. Then for \(H_{{\mathbf {c}},\tilde{s}} = \{Z \in \mathfrak {H}_g : Z {\mathbf {c}} = \tilde{s} {\mathbf {c}} \} \subset \mathfrak {H}_g\), we have

$$\begin{aligned} {\mathrm{codim}}_{\tilde{S}^{\mathrm {biZar}}}(H_{{\mathbf {c}},\tilde{s}} \cap \tilde{S}^{\mathrm {biZar}}) = g. \end{aligned}$$
(10.4)

Now that \(\tilde{S}^{\mathrm {biZar}}\) is affine linear in \(\mathfrak {H}_g\) by Lemma 10.2.4. So for such a \({\mathbf {c}}\), (10.4) holds for any \(\tilde{s} \in \tilde{S}\) because \(H_{{\mathbf {c}},\tilde{s}}\) is also affine linear in \(\mathfrak {H}_g\). Hence we may assume that \(\tilde{s}\) is Hodge generic in \(\tilde{S}\), namely \(\mathrm {MT}(\tilde{s}) = \mathrm {MT}(S)\).

Applying Condition ACZ to this \(\tilde{s}\) and \({\mathbf {c}}\), we obtain a complex analytic variety \(\tilde{C}\) of dimension \(\dim S - g + 1\) passing through \(\tilde{s}\) such that \(\tilde{C} \subset \tilde{S} \cap H_{{\mathbf {c}},\tilde{s}}\). Then (10.4) implies

$$\begin{aligned} \dim \tilde{C}^{{Zar}} \le \dim (H_{{\mathbf {c}},\tilde{s}} \cap \tilde{S}^{\mathrm {biZar}}) = \dim \tilde{S}^{\mathrm {biZar}} - g. \end{aligned}$$
(10.5)

On the other hand \(\tilde{C} \subset \tilde{S}\), so \({u}(\tilde{C}) \subset {u}(\tilde{S}) = S\). Hence

$$\begin{aligned} \dim {u}(\tilde{C})^{{Zar}} \le \dim S. \end{aligned}$$
(10.6)

Apply Ax-Schanuel, namely Theorem 10.2.5, to \(\tilde{C}\). We obtain

$$\begin{aligned}&\dim \tilde{C}^{{Zar}} + \dim {u}(\tilde{C})^{{Zar}} \ge \dim \tilde{C} + \dim \tilde{C}^{biZar}\nonumber \\&\quad = \dim S - g + 1 + \dim \tilde{C}^{\mathrm {biZar}}. \end{aligned}$$
(10.7)

By (10.5), (10.6) and (10.7), we get \( \dim \tilde{S}^{\mathrm {biZar}} > \dim \tilde{C}^{\mathrm {biZar}}. \) Thus in order to get a contradiction, it suffice to prove \(\tilde{S}^{\mathrm {biZar}} = \tilde{C}^{\mathrm {biZar}}\). We shall use condition (i) of Theorem 10.1.3 to prove this fact.

The logarithmic Ax theorem for \({{\mathcal {A}}}_g\) says that \(\tilde{S}^{\mathrm {biZar}} = H_S^{\circ }({\mathbb {R}})^+\tilde{s}\). We refer to [11, Theorem 8.1] for this theorem. Recall that \(\tilde{s}\) is Hodge generic in \(\tilde{S}\). Hence \(H_S^{\circ }\) is normal in \(\mathrm {MT}(\tilde{s})\) by André [2, Theorem 1].

By Ullmo–Yafaev [34, Theorem 1.2], bi-algebraic subsets of \(\mathfrak {H}_g\) are precisely the weakly special subsets of \(\mathfrak {H}_g\). Now \(\tilde{C}^{\mathrm {biZar}}\) contains \(\tilde{s}\) which is Hodge generic in \(\tilde{S}\), and \(\tilde{C}^{\mathrm {biZar}} \subset \tilde{S}^{\mathrm {biZar}} \subset \mathrm {MT}(\tilde{s})({\mathbb {R}})^+\tilde{s}\). So by definition of weakly special subvarieties (see [34, Definition 2.1] or [29, Definition 4.1.(b)]), we have \(\tilde{C}^{\mathrm {biZar}} = N({\mathbb {R}})^+\tilde{s}\) for some normal subgroup N of \(\mathrm {MT}(\tilde{s})\). Now \(N({\mathbb {R}})^+\tilde{s} \subset H_S^{\circ }({\mathbb {R}})^+\tilde{s}\), both N and \(H_S^{\circ }\) are normal subgroups of the reductive group \(\mathrm {MT}(\tilde{s})\), and \(H_S^{\circ }\) is simple by condition (i) of Theorem 10.1.3. Hence \(N({\mathbb {R}})^+\tilde{s} = H_S^{\circ }({\mathbb {R}})^+\tilde{s}\), and so \(\tilde{C}^{\mathrm {biZar}} = \tilde{S}^{\mathrm {biZar}}\). \(\square \)

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André, Y., Corvaja, P. & Zannier, U. The Betti map associated to a section of an abelian scheme. Invent. math. 222, 161–202 (2020). https://doi.org/10.1007/s00222-020-00963-w

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