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

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.