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

TL;DR

This paper investigates unlikely intersections in families of CM abelian varieties, extending previous results by analyzing intersections with algebraic subgroups and providing explicit bounds for canonical heights under endomorphisms.

Contribution

It generalizes prior work on unlikely intersections by considering CM fibers and offers explicit height bounds under endomorphisms for abelian varieties.

Findings

Finiteness of intersections with algebraic subgroups in CM fibers.

Explicit bounds for canonical heights under endomorphisms.

Extension of previous results to more general abelian schemes.

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 $\lambda \in \mathbb{R}$ for which the divisors $\lambda D - f^* D$ and $f^* D - \lambda D$ are ample.