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