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Finite translation orbits on double families of abelian varieties (with an appendix by E. Amerik)

Ekaterina Amerik, Paolo Dolce, Francesco Tropeano

TL;DR

This work extends the study of finite-orbit points under translations induced by a non-torsion section from single abelian-fibration settings to dual g-dimensional fibrations over bases of dimension at most g. It develops a higher-dimensional Pila–Zannier framework leveraging the Betti map, o-minimality, and a height-inequality toolkit to constrain the torsion behavior across two interlinked fibrations. The main result shows that the finite-orbit locus is contained in the union of the inverse images of two proper Zariski-closed sets Z1⊂S1 and Z2⊂S2, generalizing relative Manin–Mumford phenomena to double abelian fibrations. The work combines height bounds, conjugate-point control, and sophisticated definable-set counting to obtain uniform torsion bounds and deduce finiteness of orbits, with explicit geometric consequences and potential for further generalizations to more fibration factors.

Abstract

We study two families of $g$-dimensional abelian varieties, induced by distinct rational maps defined on a common variety $\overline{\mathcal A}$ and mapping to two bases $\overline{S}_1$ and $\overline{S}_2$. Two non-torsion sections induce birational fiberwise translations on $\overline{\mathcal A}$. We consider the action of a specific subset of the group generated by these translations. Under the assumption that $\operatorname{dim} \overline{S}_1 (= \operatorname{dim} \overline{S}_2) \leq g$, we prove that the points with finite orbit are contained in a proper Zariski closed subset. This subset is explicitly described to a certain extent. Our results generalize a theorem of Corvaja, Tsimermann, and Zannier to higher dimensions.

Finite translation orbits on double families of abelian varieties (with an appendix by E. Amerik)

TL;DR

This work extends the study of finite-orbit points under translations induced by a non-torsion section from single abelian-fibration settings to dual g-dimensional fibrations over bases of dimension at most g. It develops a higher-dimensional Pila–Zannier framework leveraging the Betti map, o-minimality, and a height-inequality toolkit to constrain the torsion behavior across two interlinked fibrations. The main result shows that the finite-orbit locus is contained in the union of the inverse images of two proper Zariski-closed sets Z1⊂S1 and Z2⊂S2, generalizing relative Manin–Mumford phenomena to double abelian fibrations. The work combines height bounds, conjugate-point control, and sophisticated definable-set counting to obtain uniform torsion bounds and deduce finiteness of orbits, with explicit geometric consequences and potential for further generalizations to more fibration factors.

Abstract

We study two families of -dimensional abelian varieties, induced by distinct rational maps defined on a common variety and mapping to two bases and . Two non-torsion sections induce birational fiberwise translations on . We consider the action of a specific subset of the group generated by these translations. Under the assumption that , we prove that the points with finite orbit are contained in a proper Zariski closed subset. This subset is explicitly described to a certain extent. Our results generalize a theorem of Corvaja, Tsimermann, and Zannier to higher dimensions.
Paper Structure (23 sections, 15 theorems, 110 equations, 2 figures)

This paper contains 23 sections, 15 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General notations.
  3. Acknowledgements
  4. Auxiliary results
  5. Betti map
  6. Height bounds
  7. Torsion bounds
  8. Control on conjugate points
  9. The main theorem
  10. Setup of the proof
  11. Construction of the heights
  12. Removing Zariski closed subsets
  13. Uniform bounds
  14. Removing euclidean open subsets
  15. Auxiliary families of abelian schemes
  16. ...and 8 more sections

Key Result

Theorem 4

Let $f_1:\overline{\mathcal{A}}\dashrightarrow \overline{S}_1$ and $f_2:\overline{\mathcal{A}}\dashrightarrow \overline{S}_2$ be a double $g$-dimensional abelian rational fibration with $\overline{\mathcal{A}}$, $\overline S_1$ and $\overline S_2$ projective varieties. Moreover, assume that $f_1$ an

Figures (2)

  • Figure 1: A representation of the portion of conjugates of $\alpha$ that stay away from a euclidean open set $W_{i,\delta}$ that tightly encircles a Zariski closed set $B$. The set $U_i$ is a selected affine chart.
  • Figure 2: A schematization of the family $\mathcal{X}\to T_{b,\delta}$.

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 10
  • Remark 1.1
  • Proposition 1.2
  • ...and 20 more