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Effective isotrivial Mordell-Lang in positive characteristic

Jason Bell, Dragos Ghioca, Rahim Moosa

Abstract

The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\capΓ$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $Γ$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $Γ$. The approach is to use finite automata to give a concrete description of $X\cap Γ$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $Γ=G(K)$. As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in $X(K)$ of bounded height. As an application of the effective description of $X\capΓ$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset?

Effective isotrivial Mordell-Lang in positive characteristic

Abstract

The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set when is a subvariety of a semiabelian variety over a finite field and is a finitely generated subgroup of that is invariant under the -power Frobenius endomorphism . That description is here made effective, and extended to arbitrary commutative algebraic groups and arbitrary finitely generated -submodules . The approach is to use finite automata to give a concrete description of . These methods and results have new applications even when specialised to the case when is an abelian variety over a finite field, a subvariety defined over a function field , and . As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in of bounded height. As an application of the effective description of , decision procedures are given for the following three diophantine problems: Is nonempty? Is it infinite? Does it contain an infinite coset?

Paper Structure

This paper contains 21 sections, 36 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Mordell-Lang for finitely generated subgroups of isotrivial commutative algebraic groups
  3. $F$-automaticity
  4. Explicit weak spanning sets
  5. A finiteness result on Frobenius pullbacks
  6. Proposition \ref{['bound']} is effective
  7. An automaton recognising $X\cap\Gamma$
  8. Effectivity
  9. Deciding rational points on subvarieties of isotrivial abelian varieties
  10. Finding generators for $G(K)$
  11. Whether or not $X(K)$ is empty is decidable.
  12. Whether or not $X(K)$ is infinite is decidable.
  13. Finding an actual spanning set for $G(K)$
  14. Whether or not $X(K)$ contains an infinite coset is decidable.
  15. A gap theorem for rational points of bounded height on subvarieties of isotrivial abelian varieties
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose $G$ is a semiabelian variety over a finite field $\mathbb F_q$ of prime characteristic $p$, and let $F:G\to G$ be the endomorphism induced by the $q$-power Frobenius. Suppose $X\subseteq G$ is a closed subvariety defined over a field extension of $\mathbb F_q$, and $\Gamma\leq G$ is a finite

Theorems & Definitions (96)

  • Theorem 1.1: Moosa-Scanlon fsets
  • Theorem A
  • Corollary 1.2
  • Corollary 1.3
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 86 more