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A geometric approach to the uniform boundedness of $\ell$-primary torsion points

Zhuchao Ji, Jiarui Song, Junyi Xie

TL;DR

This work delivers a geometric proof of the uniform boundedness of $\ell$-primary torsion points on fibers of an abelian scheme over a smooth curve by leveraging Betti foliations and an arithmetic equidistribution theorem. The core advance is a criterion: a generic, small-height sequence in the generic fiber yields isotriviality of the whole family when the genus of Zariski closures grows sublinearly with the degree. From this, the authors obtain a new proof of Cadoret–Tamagawa’s uniform boundedness result, and they resolve their conjecture without extra assumptions, while also establishing a Bogomolov-type finiteness principle and related consequences for torsion and sections. The approach marries complex-analytic dynamics (Betti foliations and monodromy) with arithmetic geometry (adelic heights and equidistribution) to produce robust geometric control over torsion phenomena in families of abelian varieties.

Abstract

We prove that for a non-isotrivial abelian scheme over a smooth curve, the genus of a generic sequence of multi-sections with small heights tends to infinity. As an application, we give a new proof of the uniform boundedness of $\ell$-primary torsion points on fibers of an abelian scheme over a smooth curve, a result originally proved by Cadoret and Tamagawa. Furthermore, our approach allows us to resolve a conjecture of Cadoret and Tamagawa without additional assumptions. Our approach is based on the theory of Betti foliations and the arithmetic equidistribution theorem.

A geometric approach to the uniform boundedness of $\ell$-primary torsion points

TL;DR

This work delivers a geometric proof of the uniform boundedness of -primary torsion points on fibers of an abelian scheme over a smooth curve by leveraging Betti foliations and an arithmetic equidistribution theorem. The core advance is a criterion: a generic, small-height sequence in the generic fiber yields isotriviality of the whole family when the genus of Zariski closures grows sublinearly with the degree. From this, the authors obtain a new proof of Cadoret–Tamagawa’s uniform boundedness result, and they resolve their conjecture without extra assumptions, while also establishing a Bogomolov-type finiteness principle and related consequences for torsion and sections. The approach marries complex-analytic dynamics (Betti foliations and monodromy) with arithmetic geometry (adelic heights and equidistribution) to produce robust geometric control over torsion phenomena in families of abelian varieties.

Abstract

We prove that for a non-isotrivial abelian scheme over a smooth curve, the genus of a generic sequence of multi-sections with small heights tends to infinity. As an application, we give a new proof of the uniform boundedness of -primary torsion points on fibers of an abelian scheme over a smooth curve, a result originally proved by Cadoret and Tamagawa. Furthermore, our approach allows us to resolve a conjecture of Cadoret and Tamagawa without additional assumptions. Our approach is based on the theory of Betti foliations and the arithmetic equidistribution theorem.
Paper Structure (12 sections, 27 theorems, 177 equations)

This paper contains 12 sections, 27 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Ideas of proofs
  4. Notations and Terminology
  5. Preliminaries
  6. The moduli space of abelian varieties
  7. The Betti foliation
  8. The Néron--Tate height
  9. The equidistribution theorem
  10. Other lemmas
  11. Proof of Theorem \ref{['thm_main1']}
  12. Applications to the uniform boundedness conjecture

Key Result

Theorem 1.2

Let $k$ be a finitely generated field over $\mathbb{Q}$, $B$ be a smooth curve over $k$, and $\pi: \mathcal{A} \to B$ be an abelian scheme. For any finite extension $k'/k$ and any integer $\ell$, there exists an integer $N \colonequals N(\mathcal{A}, B, k, k', \ell)$ such that for all $b \in B(k')$ where $\mathcal{A}_b(k')[\ell^{\infty}] \colonequals \bigcup\limits_{n=0}^{\infty} \mathcal{A}_b(k

Theorems & Definitions (65)

  • Conjecture 1.1
  • Theorem 1.2: CT12
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Example 1.7
  • Conjecture 1.8
  • Definition 2.1
  • Proposition 2.2
  • ...and 55 more