Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds

Rikuto Ito

Paper

Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds

Abstract

We construct an arithmetic period map for cubic fourfolds, in direct analogy with Rizov's work on K3 surfaces. For each

, we introduce a Deligne-Mumford stack

of cubic fourfolds with level structure and prove that the associated period map

is algebraic, étale, and descends to

whenever

is coprime to

. As an application, we develop complex multiplication theory for cubic fourfolds and show that every cubic fourfold of CM type is defined over an abelian extension of its reflex field. Moreover, using the CM theory for rank-21 cubic fourfolds, we provide an alternative proof of the modularity of rank-21 cubic fourfolds established by Livné.