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New gap principle for semiabelian varieties using globally valued fields

Nuno Hultberg

TL;DR

The paper extends the Gao-Ge-Kühne gap principle from abelian varieties to semiabelian varieties over globally valued fields (GVFs), by reducing the Bogomolov conjecture (BC) for semiabelians to BC for abelian varieties over GVFs. It leverages ultrafilter degenerations, model-theoretic compactness, and moduli spaces to transfer small-point/height statements from the abelian setting to the semiabelian setting, and proves BC for abelian varieties over GVFs of characteristic $0$, with unconditional positive-characteristic results when the abelian quotient is an elliptic curve. Consequently, it derives a new gap principle for semiabelian varieties, including a characteristic-free formulation in the elliptic-quotient case and partial results in positive characteristic. The work unifies Arakelov-style intersection theory on GVFs with model-theoretic and dynamical perspectives, enabling uniform height bounds and a pathway to uniform Mordell-Lang-type statements in families. Significantly, it shows that BCSA follows from BC for abelian varieties over GVFs, and provides a robust framework for transferring height-density phenomena across GVFs via ultraproducts and moduli.

Abstract

Hrushovski observed that the new gap principle of Gao-Ge-Kühne is essentially equivalent to the Bogomolov conjecture over arbitrary globally valued fields of characteristic $0$. Building on this observation, we prove a new gap principle for semiabelian varieties by reducing the Bogomolov conjecture for semiabelian varieties to the Bogomolov conjecture for abelian varieties over arbitrary GVFs. This reduction remains valid in positive characteristic; however, the corresponding Bogomolov conjecture for abelian varieties is not yet known in that setting. We prove an unconditional new gap principle in positive characteristic for semiabelian varieties whose abelian quotient is an elliptic curve.

New gap principle for semiabelian varieties using globally valued fields

TL;DR

The paper extends the Gao-Ge-Kühne gap principle from abelian varieties to semiabelian varieties over globally valued fields (GVFs), by reducing the Bogomolov conjecture (BC) for semiabelians to BC for abelian varieties over GVFs. It leverages ultrafilter degenerations, model-theoretic compactness, and moduli spaces to transfer small-point/height statements from the abelian setting to the semiabelian setting, and proves BC for abelian varieties over GVFs of characteristic , with unconditional positive-characteristic results when the abelian quotient is an elliptic curve. Consequently, it derives a new gap principle for semiabelian varieties, including a characteristic-free formulation in the elliptic-quotient case and partial results in positive characteristic. The work unifies Arakelov-style intersection theory on GVFs with model-theoretic and dynamical perspectives, enabling uniform height bounds and a pathway to uniform Mordell-Lang-type statements in families. Significantly, it shows that BCSA follows from BC for abelian varieties over GVFs, and provides a robust framework for transferring height-density phenomena across GVFs via ultraproducts and moduli.

Abstract

Hrushovski observed that the new gap principle of Gao-Ge-Kühne is essentially equivalent to the Bogomolov conjecture over arbitrary globally valued fields of characteristic . Building on this observation, we prove a new gap principle for semiabelian varieties by reducing the Bogomolov conjecture for semiabelian varieties to the Bogomolov conjecture for abelian varieties over arbitrary GVFs. This reduction remains valid in positive characteristic; however, the corresponding Bogomolov conjecture for abelian varieties is not yet known in that setting. We prove an unconditional new gap principle in positive characteristic for semiabelian varieties whose abelian quotient is an elliptic curve.
Paper Structure (22 sections, 57 theorems, 80 equations)

This paper contains 22 sections, 57 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries on globally valued fields
  4. Definitions of globally valued fields
  5. Intersection numbers over GVFs
  6. Polarized GVF structures
  7. Ultraproducts of GVFs
  8. GVF analytification and existential closedness
  9. Families of subvarieties and Zhang inequalities
  10. Bogomolov conjecture
  11. Preliminaries on semiabelian varieties
  12. Statements and equivalent formulations
  13. New gap principle
  14. Moduli of semiabelian varieties
  15. Reduction to the Bogomolov conjecture
  16. ...and 7 more sections

Key Result

Theorem 1

$\mathrm{(BC)}$ is true for every abelian variety over a GVF of characteristic $0$.

Theorems & Definitions (132)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3: New gap principle over $\Qbar$
  • Theorem 4: New gap principle in big characteristic
  • Theorem 5
  • Proposition 2: Proposition \ref{['prop:lower_zhang_ineq']} and \ref{['prop:zhang_inequality']}
  • Theorem 6: Corollary \ref{['cor:uniformzhang']}
  • Definition 2.1.1
  • Lemma 2.1.2
  • ...and 122 more