Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds

TL;DR

This paper develops an arithmetic period map for cubic fourfolds, establishing algebraic and étale properties, and applies complex multiplication theory to show CM cubic fourfolds are defined over abelian extensions, also providing an alternative proof of their modularity.

Contribution

It introduces a new arithmetic period map for cubic fourfolds, extending complex multiplication theory and proving modularity for rank-21 cases.

Findings

Constructed algebraic, étale period map for cubic fourfolds

Proved CM cubic fourfolds are defined over abelian extensions

Provided an alternative proof of modularity for rank-21 cubic fourfolds

Abstract

We construct an arithmetic period map for cubic fourfolds, in direct analogy with Rizov's work on K3 surfaces. For each $N\geq 1$, we introduce a Deligne-Mumford stack $\widetilde{\mathcal{C}^{[N]}}$ of cubic fourfolds with level structure and prove that the associated period map $j_{N}:\widetilde{\mathcal{C}^{[N]}}_{\mathbb{C}}\to {\rm Sh}_{K_{N}}(L)_{\mathbb{C}}$ is algebraic, \'etale, and descends to $\mathbb{Q}$ whenever $N$ is coprime to $2310$. 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 Livn\'e.