Heights and dynamics over arbitrary fields

Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and Yuan-Zhang, we show how to define arithmetic intersections and heights relative to any finitely generated field extension K/kK/k, and construct canonical heights for polarizable arithmetic dynamical systems f:X→Xf:X→X. These heights have a corresponding Northcott property when kk is ℚQ or FqFq. When kk is larger, we show that Northcott for canonical heights is conditional on the non-isotriviality of f:X→Xf:X→X, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. Additionally, we prove the Hodge Index Theorem for arithmetic intersections relative to K/kK/k. Since, when Northcott holds, preperiodic points are the same as height zero points, this has applications to dynamical systems. By the Lefschetz principle, these results can be applied over any field.