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Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms

Luca Ferrigno

TL;DR

The paper addresses finiteness of unlikely intersections between an irreducible curve in a non-isotrivial abelian scheme and the CM-subgroup structure of CM fibers. It develops explicit, optimal bounds for canonical heights under endomorphisms of abelian varieties, expressed through the eigenvalues of the Rosati-rotated endomorphism $f^{\dagger}f$, and extends these to homomorphisms between abelian varieties. Building on the Pila–Zannier framework and o-minimality, the author reduces to the universal family, controls definable sets, and uses height inequalities to bound CM-related intersections, culminating in a finite intersection result (a CM-variant of Barroero–Capuano). The work thus connects explicit height theory with unlikely-intersection heuristics, providing effective constants and advancing the Zilber–Pink landscape for families of abelian varieties.

Abstract

Let $S$ be a smooth irreducible curve over $\overline{\mathbb{Q}}$, and let $\mathcal{A} \to S$ be an abelian scheme with a curve $C \subset \mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In 2020, Barroero and Capuano proved that if $C$ is not contained in a proper subgroup scheme, then the intersection of $C$ with the union of the flat subgroup schemes of $\mathcal{A}$ of codimension at least 2 is finite. In this article, we continue to study this problem by considering the intersections with the algebraic subgroups of the CM fibers, generalizing a previous result of Barroero for fibered powers of elliptic schemes. A key ingredient of the proof is an explicit control of canonical heights under endomorphisms: for an abelian variety $A/\overline{\mathbb{Q}}$, an ample symmetric divisor $D$, and $f \in \mathrm{End}(A)$, we bound explicitly $\widehat{h}_{A, D}(f(P))$ in terms of $\widehat{h}_{A, D}(P)$ by determining the values of $λ\in \mathbb{R}$ for which the divisors $λD - f^* D$ and $f^* D - λD$ are ample.

Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms

TL;DR

The paper addresses finiteness of unlikely intersections between an irreducible curve in a non-isotrivial abelian scheme and the CM-subgroup structure of CM fibers. It develops explicit, optimal bounds for canonical heights under endomorphisms of abelian varieties, expressed through the eigenvalues of the Rosati-rotated endomorphism , and extends these to homomorphisms between abelian varieties. Building on the Pila–Zannier framework and o-minimality, the author reduces to the universal family, controls definable sets, and uses height inequalities to bound CM-related intersections, culminating in a finite intersection result (a CM-variant of Barroero–Capuano). The work thus connects explicit height theory with unlikely-intersection heuristics, providing effective constants and advancing the Zilber–Pink landscape for families of abelian varieties.

Abstract

Let be a smooth irreducible curve over , and let be an abelian scheme with a curve , both defined over . In 2020, Barroero and Capuano proved that if is not contained in a proper subgroup scheme, then the intersection of with the union of the flat subgroup schemes of of codimension at least 2 is finite. In this article, we continue to study this problem by considering the intersections with the algebraic subgroups of the CM fibers, generalizing a previous result of Barroero for fibered powers of elliptic schemes. A key ingredient of the proof is an explicit control of canonical heights under endomorphisms: for an abelian variety , an ample symmetric divisor , and , we bound explicitly in terms of by determining the values of for which the divisors and are ample.
Paper Structure (17 sections, 35 theorems, 170 equations)

This paper contains 17 sections, 35 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abelian varieties and their endomorphisms
  4. Moduli spaces, universal families and their uniformizations
  5. Heights
  6. Complex Multiplication
  7. Reduction to the universal family of principally polarized abelian varieties
  8. O-minimality and definable sets
  9. Matrix bounds for endomorphisms of abelian varieties
  10. The main estimate
  11. Canonical height bounds under endomorphisms
  12. Properties of endomorphisms and line bundles of abelian varieties
  13. Proof of Theorem \ref{['thm:can_heights_endomorphisms']}
  14. Height bounds for homomorphisms between abelian varieties
  15. A height inequality
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $S$ and $\mathcal{A} \rightarrow S$ be as above and assume that $\mathcal{A}$ is not isotrivial. Let $\mathcal{C} \subseteq \mathcal{A}$ an irreducible curve defined over $\overline{\mathbb{Q}}$ that is neither contained in a fixed fiber nor in a translate of a proper flat subgroup scheme of $\m

Theorems & Definitions (76)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 66 more