Finiteness of Heights in Isogeny Classes of Motives

Abstract

Using integral p-adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness property for heights in the isogeny class of a motive, where the isogenous motives are not required to be defined over the same number field. This expands on a result of Kisin and Mocz for the Faltings height in isogeny classes of abelian varieties. As a corollary, we prove a result about finiteness of $\overline{\mathbb{Q}}-points with bounded height in Hecke orbits of arbitrary Shimura varieties.