[Paper Review] Positive characteristic Manin-Mumford theorem

Thomas Scanlon|ArXiv.org|Mar 26, 2003

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a positive characteristic analogue of the Manin-Mumford conjecture for semiabelian varieties using model-theoretic methods, specifically the theory of existentially closed difference fields (ACFA). It proves that the Zariski closure of torsion points in a subvariety is a finite union of special subvarieties—defined via isogenies from algebraic groups over finite fields—thereby generalizing Raynaud's theorem to positive characteristic.

ABSTRACT

We prove a version of the Manin-Mumford conjecture for semiabelian varieties over fields of positive characteristic. The proof presented here contains the details of the proof sketched by the author in the article "Diophantine geometry from model theory," BSL 7 (2001) no. 1, 37 - 57; using the model theory of difference fields along the lines of Hrushovski. Algebraic proofs have been presented by Pink & Roessler and Pillay.

Motivation & Objective

• To establish a positive characteristic analogue of the Manin-Mumford conjecture for semiabelian varieties.
• To address the failure of the classical Manin-Mumford statement in positive characteristic by introducing a refined notion of 'special' subvarieties.
• To prove that the Zariski closure of torsion points in a subvariety is a finite union of such special subvarieties.
• To apply model-theoretic tools, particularly ACFA and the dichotomy theorem for difference fields, to resolve the problem.

Proposed method

• Utilizes the model theory of existentially closed difference fields (ACFA) to analyze torsion points in positive characteristic.
• Constructs a discrete valuation ring with finite residue field and a semiabelian model over it to lift torsion structure.
• Applies Hrushovski’s dichotomy theorem for definable groups in ACFA to classify subgroups of torsion points.
• Uses the Frobenius endomorphism on the special fiber to define a polynomial $ P(X) \in \mathbb{Z}[X] $ whose roots are not roots of unity.
• Reduces the problem to analyzing $ \ker P(\sigma) $, where $ \sigma $ is a field automorphism, and decomposes this group into modular and essentially algebraic parts.
• Applies the socle theorem and properties of isogenies to show that torsion points lie in cosets of subvarieties defined over $ \mathbb{F}_p^{\rm alg} $, hence are special.

Experimental results

Research questions

• RQ1 What is the correct analogue of the Manin-Mumford conjecture in positive characteristic?
• RQ2 How can torsion points in semiabelian varieties over fields of positive characteristic be Zariski-closed in a finite union of subvarieties?
• RQ3 What role does the Frobenius endomorphism play in defining the structure of torsion subgroups in positive characteristic?
• RQ4 Can model-theoretic tools like ACFA and the dichotomy theorem be used to prove arithmetic results in algebraic geometry?
• RQ5 What conditions define a 'special' subvariety in positive characteristic, and how do they relate to torsion points?

Key findings

• The Zariski closure of the set of torsion points in a subvariety $ X \subseteq G $, where $ G $ is a semiabelian variety over an algebraically closed field of positive characteristic, is a finite union of special subvarieties.
• Special subvarieties are defined as translates of preimages of subvarieties under isogenies from algebraic groups defined over $ \mathbb{F}_p^{\rm alg} $.
• The torsion subgroup $ G(K)_{\rm tor} $ is isomorphic to $ G(S')_{\rm tor} $, where $ S' $ is the perfection of a maximal unramified extension, ensuring torsion points are captured in a model over a finite field.
• The polynomial $ P(X) \in \mathbb{Z}[X] $, constructed from the Frobenius action on the special fiber, has no complex roots that are roots of unity, which is essential for the dichotomy argument.
• Definable subgroups of $ \ker P(\sigma) $ are either modular or essentially algebraic, and the latter are commensurable with images of algebraic groups over finite fields.
• The proof concludes that any irreducible subvariety $ X $ with Zariski-dense torsion points must be special, i.e., a translate of the preimage of a subvariety defined over $ \mathbb{F}_p^{\rm alg} $.