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

Let $S$ be a smooth irreducible curve over $\bar{\mathbb{Q}}$, and let $\mathcal{A}\to S$ be an abelian scheme with a curve $C\subset \mathcal{A}$, both defined over $\bar{\mathbb{Q}}$. 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 $A\mathrm{/}\bar{\mathbb{Q}}$, an ample symmetric divisor $D$, and $f\in \mathrm{E}\mathrm{n}\mathrm{d}(A)$, 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 $\lambda D-f^{\ast }D$ and $f^{\ast }D-\lambda D$ are ample.