A Global Crystalline Period Map

TL;DR

This paper introduces a new global construction of the crystalline period map, extending its applicability from local to global settings on the moduli stack of p-divisible groups, with implications for number theory and homotopy theory.

Contribution

It provides an alternative, global construction of the crystalline period map on the entire moduli stack of p-divisible groups, surpassing the local limitations of previous methods.

Findings

Constructs a global crystalline period map on the moduli stack of p-divisible groups.

Specializes to the original local period map in the appropriate setting.

Potential applications to Langlands correspondences and homotopy groups of spheres.

Abstract

The crystalline period map is a tool for linearizing $p$-divisible groups. It has been applied to study the Langlands correspondences, and has possible applications to the homotopy groups of spheres. The original construction of the period map is inherently local. We present an alternative construction, giving a map on the entire moduli stack of $p$-divisbile groups, up to isogeny, which specializes to the original local construction.