[Paper Review] Uniform boundedness of rational points and preperiodic points

Bjorn Poonen|arXiv (Cornell University)|Jun 29, 2012

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper investigates uniform boundedness of rational and preperiodic points across families of algebraic varieties and dynamical systems over number fields and finitely generated extensions of ℚ. It establishes equivalences between uniform boundedness questions for rational points and preperiodic points, and shows that a positive answer to the Morton–Silverman conjecture implies uniform boundedness of torsion points on abelian varieties.

ABSTRACT

We ask questions generalizing uniform versions of conjectures of Mordell and Lang and combining them with the Morton--Silverman conjecture on preperiodic points. We prove a few results relating different versions of such questions.

Motivation & Objective

• To investigate uniform boundedness of rational points across families of algebraic varieties over finitely generated extensions of ℚ.
• To generalize Mordell–Lang and Bombieri–Lang-type conjectures to higher-dimensional and higher-genus families.
• To explore analogues of uniform boundedness for preperiodic points under endomorphisms of algebraic varieties.
• To establish logical equivalences between different formulations of uniform boundedness across varying field extensions and geometric families.
• To connect the Morton–Silverman conjecture on preperiodic points to uniform boundedness of torsion points on abelian varieties.

Proposed method

• Reduces general uniform boundedness questions to the case of affine schemes via finite-type decomposition.
• Applies restrictions of scalars to relate boundedness over degree-D extensions to boundedness over the base field.
• Uses the theory of Néron models and base change to construct universal families of varieties and dynamical systems.
• Employs Skolem’s trick to reduce systems of polynomial equations to degree-4 hypersurfaces in affine space.
• Applies Zarhin’s trick to reduce the study of abelian varieties to principally polarized ones, enabling the use of versal families.
• Establishes equivalence between uniform boundedness for degree-D extensions and the base field via constructions of relative Weil restrictions.

Experimental results

Research questions

• RQ1 For each n ≥ 1, is there a uniform bound B_n on the number of rational points on any degree-4 hypersurface in 𝔸^n over ℚ with finitely many rational points?
• RQ2 For a morphism π: X → S of finite-type k-schemes over a finitely generated extension k of ℚ, is the set {#X_s(k) : s ∈ S(k)} finite when X_s(k) is finite?
• RQ3 For a fixed degree D, is the set {#PrePer(f_s, L) : [L:k] = D, s ∈ S(L)} finite, where f_s is a self-map on the fiber X_s?
• RQ4 Does the Morton–Silverman conjecture on preperiodic points imply uniform boundedness of torsion points on abelian varieties over number fields?
• RQ5 Are the uniform boundedness questions for rational points over number fields equivalent to those over arbitrary finitely generated extensions of ℚ?

Key findings

• Question 1.2 (uniform boundedness of rational points over finitely generated fields) is equivalent to Question 1.3 (bounded degree extensions) for any finitely generated extension k of ℚ.
• The uniform boundedness of rational points on degree-4 hypersurfaces in 𝔸^n over ℚ implies the uniform boundedness of rational points on all curves of genus >1 over number fields.
• A positive answer to Question 1.1 (degree-4 uniform boundedness) would imply a positive answer to the number field case of the Caporaso–Harris–Mazur conjecture.
• The Morton–Silverman conjecture implies uniform boundedness of torsion points on abelian varieties over number fields, as shown via the 2-isogeny on elliptic curves.
• Question 3.2 (uniform boundedness of preperiodic points over degree-D extensions) is equivalent to the base case D=1 when X and S are quasi-projective, via Weil restriction of scalars.
• The restriction of scalars construction ensures that uniform boundedness over the base field implies uniform boundedness over all degree-D extensions, even when X is not affine.