Joint IAS/Princeton University Number Theory Seminar

Around 1990, Joe Silverman wrote a series of three articles on the variation of canonical height in families of elliptic curves. I will discuss connections between these results and dynamical systems on $\mathbb P^1$ (and an associated Berkovich space). As the height functions define dynamical "bifurcation measures" on the base variety, I will show illustrations of these measure densities. The main new result is that the points of small height will equidistribute in the base of the family, and I will describe applications to Unlikely Intersection problems. If there is time, I will also discuss the rationality of canonical height, when working over function fields.