Breuil-Kisin-Fargues modules with complex multiplication

TL;DR

This paper establishes the Tannakian nature of Breuil-Kisin-Fargues modules up to isogeny and classifies those with complex multiplication, drawing parallels with classical Hodge theory and identifying a p-adic Serre group.

Contribution

It introduces and classifies Breuil-Kisin-Fargues modules with complex multiplication, extending classical theory into the p-adic setting.

Findings

The category of modules is Tannakian.

Classification of modules with complex multiplication.

Identification of a p-adic Serre group avatar.

Abstract

We prove that the category of (rigidified) Breuil-Kisin-Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil-Kisin-Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a "p-adic Serre group".