I will discuss canonical local heights on abelian varieties over non-archimedean fields from the point of view of Berkovich analytic spaces. Our main result is a refinement of Néron's classical result relating canonical local heights with intersection multiplicities on the Néron model. We also revisit Tate's explicit formulas for Néron's canonical local heights on elliptic curves (involving Bernoulli polynomials). Our results can be viewed as extensions of Tate's formulas to higher dimensions.

(Based on joint work with Robin de Jong.)