On period spaces for p-divisible groups

TL;DR

This paper investigates the images of period morphisms from moduli spaces of p-divisible groups to Rapoport-Zink period spaces, revealing that the images are often proper subsets, thus addressing a question posed by Grothendieck.

Contribution

It determines the images of these period morphisms and provides examples showing they are rarely surjective, advancing understanding of p-divisible groups and their associated period spaces.

Findings

Images of period morphisms are often proper subsets of the period spaces.

Surjectivity of period morphisms is rare, only in special cases.

Addresses a longstanding question of Grothendieck regarding the images.

Abstract

In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.