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Integral torsion points on abelian varieties over function fields

Robin de Jong, Nicole Looper, Farbod Shokrieh

Abstract

We prove an analogue, over global function fields, of a conjecture due to Su-Ion Ih concerning the non-Zariski density of torsion points on abelian varieties that are integral with respect to a given non-special divisor. Along the way, we establish a Tate--Voloch type theorem for abelian varieties over completions of global function fields, which allows us to obtain a logarithmic equidistribution result for Galois orbits of torsion points.

Integral torsion points on abelian varieties over function fields

Abstract

We prove an analogue, over global function fields, of a conjecture due to Su-Ion Ih concerning the non-Zariski density of torsion points on abelian varieties that are integral with respect to a given non-special divisor. Along the way, we establish a Tate--Voloch type theorem for abelian varieties over completions of global function fields, which allows us to obtain a logarithmic equidistribution result for Galois orbits of torsion points.
Paper Structure (31 sections, 34 theorems, 76 equations)

This paper contains 31 sections, 34 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Our main result
  3. Ingredients of the proof
  4. A Tate--Voloch type result
  5. Back to number fields
  6. Structure of the paper
  7. Acknowledgements
  8. Preliminaries
  9. Proximity of torsion over function fields
  10. Invariant subvarieties of abelian varieties
  11. A result of Pink and Rössler
  12. A construction of invariant subvarieties
  13. Distance functions and ultrafields
  14. Ultrafields
  15. Discreteness of torsion
  16. ...and 16 more sections

Key Result

Lemma 2.2

Let $X \subseteq A$ be a closed irreducible subvariety and let $H$ be a closed subgroup variety of $A$. Then, we have a canonical identification

Theorems & Definitions (71)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 61 more