# Seminar on Rapoport-Zink spaces

In this seminar, we want to understand (part of) the book on period spaces by Rapoport and Zink. More precisely, we will study the definition and properties of the period morphism $\breve{\pi}^1 \colon \breve{\mathcal M}^{\rm rig} \longrightarrow \breve{\mathcal F}^{\rm wa}.$ Before we come to its definition (Talk 9), we will have to take our time to understand the definitions of the source and the target, though:

The space $\breve{\mathcal M}$ is a formal scheme representing a moduli functor of $p$-divisible groups (which is very close to a natural generalization of the formal schemes we have seen in the course on formal groups), Talks 4, 5, 7.

The superscript $-^{\rm rig}$ means that we pass from a formal scheme to its ``generic fiber’‘, a ``rigid analytic space’‘, Talk 8.

The weakly admissible locus $\breve{\mathcal F}^{\rm wa}$ is another rigid analytic space which arises as a subspace of a flag variety, Talk 10.

Along the way, we will see methods to analyze the étale-local structure of the formal schemes (so-called local models, Talk 6), and at the end we will discuss the image of the period morphism, Talk 11. (If there are enough people who are interested in a continuation, then next term we could look at the uniformization results for Shimura varieties presented in the final chapter of the book.)

**ECTS points:** If applicable, the seminar counts as a Master Seminar (9 ECTS points).

**Date:** Wed, 4-6pm, S-U-3.03. We start on Wed, April 19.

**Program:** pdf

We are still looking for a few more speakers – if you are interested, please contact me.

**Contact:** Ulrich Görtz, ulrich.goertz@uni-due.de

## References

S. Bosch, Lectures on Formal and Rigid Geometry, Springer Lect. Notes in Math. 2105, 2014.

J.-F. Boutot, H. Carayol, Uniformisation $p$-adique de courbes de Shimura: les théorèmes de Cerednik et Drinfeld, in: Courbes modulaires et courbes de Shimura, Astérisque 196—197, 1997. (Translation to English)

B. Conrad, Several approaches to non-archimedean geometry, AWS Lecture notes.

J.-F. Dat, S. Orlik, M. Rapoport, Period domains over finite and $p$-adic fields, Cambridge Univ. Press, 2010.

L. Fargues, “An introduction to Lubin-Tate spaces and $p$-divisible groups:https://webusers.imj-prg.fr/~laurent.fargues/Cours_Chine.dvi

G. Pappas, M. Rapoport, B. Smithling, Local models of Shimura varieties I. Geometry and combinatorics, in: Handbook of moduli (eds. G. Farkas and I. Morrison), vol. III, 135—217, Adv. Lect. in Math. 26, International Press, 2013.

M. Rapoport, Non-Archimedean Period Domains, Proc. of the International Congress of Mathematicians (Zürich, 1994), 423—434, Birkhäuser, 1995.

M. Rapoport, Accessible and weakly accessible period domains, Appendix to: P. Scholze, On the $p$-adic cohomology of the Lubin-Tate tower, arXiv:1506.04022, 2015.

M. Rapoport, Th. Zink, Period Spaces for $p$-divisible Groups, Annals of Math. Studies, Princeton Univ. Press, 1996.

H. Wang, Moduli spaces of $p$-divisible groups and period morphisms, Master’s thesis Univ. Leiden, 2009.

J. Weinstein, The Geometry of Lubin-Tate spaces

## Talks

1. | Introduction | Ulrich Görtz |

2. | $p$-divisible groups | Tobias Kreutz |

3. | Crystals and Grothendieck-Messing theory | Qijun Yan |

4. | Moduli of quasi-isogenies | Sebastian Bartling |

5. | Rapoport-Zink spaces | Shen-Ning Tung |

6. | Local models | Felix Gora |

7. | Examples: The Lubin-Tate case and the Drinfeld case | Mihir Sheth |

8. | Rigid Geometry | Heer Zhao |

9. | The period morphism | Matti Wuerthen |

10. | The image of the period morphism | Andrea Marrama |

11. | The admissible locus | Lennart Gehrmann |