Heights and arithmetic dynamics over finitely generated fields

Abstract

We develop a theory of vector-valued heights and intersections defined relative to finitely generated extensions K/k. These generalize both number field and geometric heights. When k is Q or Fp, or when a non-isotriviality condition holds, we obtain Northcott-type results. We then prove a version of the Hodge Index Theorem for vector-valued intersections, and use it to prove a rigidity theorem for polarized dynamical systems over any field.(This is proven by 41] when K has characteristic zero and by the author [7] when K is a transcendence degree one function field. We additionally prove a corollary for local fields.Totally non-isotrivial is a necessary condition which is slightly stronger than not isotrivial, and equivalent only for curves. This extends results of Baker [2] and Benedetto [3] for curves, and Lang-Néron [23] for abelian varieties.Consider a polarizable dynamical system f : X → X defined over C(s, t). By the Lefschetz principle, we may assume this system is defined over some potentially-verylarge-but-finite transcendence degree extension of Q; indeed this is how Theorem A applies to all fields K. Theorem C establishes, however, that heights relative to C(s, t)/C are sufficient, provided the dynamical system is totally non-isotrivial over C, and conversely, that the failure of Northcott finiteness for this height indicates some amount of isotriviality. In practice, these heights may be much more natural and practical to work with.We also show how to compare different relative arithmetic settings. Suppose K/k 1 /k is a tower of finitely generated extensions. Heights relative to K/k 1 can be obtained as specializations of those relative to K/k by intersecting with vertical fibers of a model for K over a model for k 1 , both over Spec k.These results have immediate applications to the field of unlikely intersections in arithmetic dynamics. Problems in this area typically study intersections of families of subvarieties, often defined to answer a dynamical question, and posit that when the intersection behaves differently than the generic case, there must be an arithmetic explanation. See [43] for an overview of problems in this area. Results are often restricted to number fields or to one dimensional families, since they rely on the use of height functions with Northcott finiteness.In some cases, one can extend results to a larger transcendence degree extension K/k, i.e. a higher dimensional family, by either specializing down from K via transcendence degree one subfields, or by building a tower of transcendence degree one extensions up from k. But both require new arguments to connect each subsequent extension and to handle isotriviality, and do not always work.Instead, using the heights and accompanying Northcott properties of this paper, one can now generalize many of these results to higher dimensional families with little additional alteration to the method. Theorem C also answers the isotriviality questions that often come up in these contexts. As an example, we can obtain the Bogomolov-type resul...