CM p-divisible Groups over Finite Fields.

Abstract

The primary motivation for this dissertation is to further the study of CM lifting of abelian varieties based on work by Chai, Conrad, and Oort. One general area of inquiry is how much control we have over the ramification of the discrete valuation ring over which the given CM abelian variety has a CM-lift after an isogeny. Questions of this kind may be solvable by translating them into questions involving $p$-divisible groups, and then into questions involving Dieudonne modules. In order to make use of this transformation, we must first classify Dieudonne modules of CM p-divisible groups over finite fields up to isomorphism and isogeny. We begin that process in this dissertation, identifying invariants which determine the isomorphism and isogeny classes of such modules.