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The isotrivial case in the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic

Dragos Ghioca

TL;DR

This paper advances the Mordell–Lang program in positive characteristic by giving an explicit, intrinsic description of X(K) ∩ Γ for semiabelian varieties G defined over a finite field, where Γ ⊂ G(K) is finitely generated. It first proves a general result (Theorem thm:G) that the intersection of an S-arithmetic set with a finitely generated subgroup is a finite union of S-arithmetic sets, achieved via reductions to a torsion-free, finitely generated ambient group and a linear-algebra reformulation that separates C- and L-subset analyses. It then deduces the main statement (Theorem thm:main) that X(K) ∩ Γ is a finite union of S_F-arithmetic sets, aligning with the Moosa–Scanlon F-set framework but extending to arbitrary Γ. The approach leverages preperiodicity of linear recurrence sequences, Laurent’s theorem on S-unit type equations, and a careful handling of recurrence roots within a powers-closed set S_F, yielding an explicit, intrinsic description of intersections in positive characteristic with potential implications for arithmetic geometry over function fields and related Diophantine problems.

Abstract

Let G be a semiabelian variety defined over a finite subfield of an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of G(K).

The isotrivial case in the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic

TL;DR

This paper advances the Mordell–Lang program in positive characteristic by giving an explicit, intrinsic description of X(K) ∩ Γ for semiabelian varieties G defined over a finite field, where Γ ⊂ G(K) is finitely generated. It first proves a general result (Theorem thm:G) that the intersection of an S-arithmetic set with a finitely generated subgroup is a finite union of S-arithmetic sets, achieved via reductions to a torsion-free, finitely generated ambient group and a linear-algebra reformulation that separates C- and L-subset analyses. It then deduces the main statement (Theorem thm:main) that X(K) ∩ Γ is a finite union of S_F-arithmetic sets, aligning with the Moosa–Scanlon F-set framework but extending to arbitrary Γ. The approach leverages preperiodicity of linear recurrence sequences, Laurent’s theorem on S-unit type equations, and a careful handling of recurrence roots within a powers-closed set S_F, yielding an explicit, intrinsic description of intersections in positive characteristic with potential implications for arithmetic geometry over function fields and related Diophantine problems.

Abstract

Let G be a semiabelian variety defined over a finite subfield of an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of G(K).
Paper Structure (13 sections, 4 theorems, 57 equations)

This paper contains 13 sections, 4 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. The intersection of a subvariety of a semiabelian variety defined over a finite field with a finitely generated subgroup invariant under a power of the Frobenius endomorphism
  3. The intersection of a subvariety of a semiabelian variety with an arbitrary finitely generated subgroup
  4. Plan for our paper
  5. Proof of Theorem \ref{['thm:G']}
  6. Plan for our proof of Theorem \ref{['thm:G']}
  7. It suffices to prove Theorem \ref{['thm:G']} assuming $\Gamma_0=\tilde{\Gamma}$ is finitely generated
  8. It suffices to prove Theorem \ref{['thm:G']} assuming $\Gamma_0$ is torsion-free
  9. Reduction to a linear algebra question
  10. The case of ${\bf C}$-subsets of ${\mathbb Z}^r$
  11. The case of ${\bf L}$-subsets of ${\mathbb Z}^r$
  12. Conclusion for our proof of Theorem \ref{['thm:G']}
  13. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.3

Let $G$, $K$, $\mathbb F_q$ and $F$ be as in Notation not:semi. Let $X\subseteq G$ be a subvariety defined over $K$ and let $\Gamma\subset G(K)$ be a finitely generated subgroup with the property that there exists $\ell\in{\mathbb N}$ such that $F^\ell(\Gamma)\subseteq \Gamma$. Then $X(K)\cap\Gamma$

Theorems & Definitions (14)

  • Definition 1.2
  • Theorem 1.3: Moosa-Scanlon F-sets
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Lemma 2.1
  • proof
  • ...and 4 more