Complex Hodge theory equips the rational singular cohomology of smooth projective variety with a Hodge filtration, and period maps measure how these filtrations vary in families. Period maps are highly transcendental in nature, and the transcendence properties of these maps reflect interesting aspects of the geometry; for example a theorem of Cohen and Shiga-Wolfhart show that a complex abelian variety A has CM if and only both A and the Hodge filtration are defined over a number field.

In this talk, we describe a similar situation in p-adic Hodge theory, and discuss joint work with Sean Howe that proves an analogue of the theorem of Cohen and Shiga-Wolfhart. The first part of the talk will contain a brief introduction to Hodge theory, so no previous experience with Hodge theory (complex or p-adic) is required!