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The Uniform Mordell-Lang Conjecture

Ziyang Gao, Tangli Ge, Lars Kühne

Abstract

The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety $X$ with a subgroup of finite rank is contained in a finite union of cosets contained in $X$. In this article, we prove a uniform version of this conjecture, meaning that that the number of cosets necessary does not depend on the ambient abelian variety. To achieve this, we prove a general gap principle on algebraic points that extends the gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov--Gao--Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.

The Uniform Mordell-Lang Conjecture

Abstract

The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety with a subgroup of finite rank is contained in a finite union of cosets contained in . In this article, we prove a uniform version of this conjecture, meaning that that the number of cosets necessary does not depend on the ambient abelian variety. To achieve this, we prove a general gap principle on algebraic points that extends the gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov--Gao--Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.

Paper Structure

This paper contains 32 sections, 30 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. The Ueno locus and first reductions
  3. A generalized gap principle
  4. Ideas of the proof
  5. Outline of the article
  6. Preliminaries on abelian varieties
  7. Polarizations
  8. Degree estimates
  9. Hilbert schemes and non-degeneracy
  10. Abelian variety
  11. Family version
  12. Non-degeneracy
  13. Applying the height inequality
  14. A result in family
  15. Proposition \ref{['PropHtIneqStep1']} for principally polarized abelian varieties
  16. ...and 17 more sections

Key Result

Theorem 1.1

For all integers $g,d \geq 0$, there exists a constant $c(g,d)>0$ with the following property. Let $X \subseteq A$ be an irreducible closed subvariety and $\Gamma\subseteq A(F)$ a subgroup of finite rank. Then the intersection $X(F)\cap\Gamma$ is covered by at most cosets contained in $X$.

Theorems & Definitions (56)

  • Theorem 1.1: Uniform Mordell--Lang Conjecture
  • Theorem \ref{MainThm2}$'$
  • Theorem \ref{MainThm2}$'$: New Gap Principle
  • Theorem \ref{MainThm2}$'$: Uniform Bogomolov Conjecture
  • Lemma \ref{MainThm2}$'$
  • Lemma \ref{MainThm2}$'$
  • proof
  • Lemma \ref{MainThm2}$'$
  • proof
  • Lemma \ref{MainThm2}$'$
  • ...and 46 more