We discuss Néron's canonical local heights on abelian varieties over non-archimedean valued fields from the point of view of Berkovich analytic spaces. We present a refinement of Néron's classical result relating his local heights with intersection multiplicities on the Néron model. Our result yields an extension to higher dimensions of Tate's explicit formulas for the canonical local heights on elliptic curves. This is based on joint work with Farbod Shokrieh.

CM points over Shimura curves can be related to periods of a rigid analytic space using the theory of p-adic uniformization. In this talk, we will introduce the key points of this relationship and study a setting where the inverse process can be made explicit. More specifically, following the work of Bertolini, Darmon, and Green, we will show that some periods defined by integrals over products of non-Archimedean upper half-planes are related to CM points of Shimura curves.