1. Introduction & Purposes

Shimura varieties are one of main interests in arithmetic geometry and representation theory.

Quaternionic Shimura varieties serve a special but very important class of Shimura varieties.

Special cases of them – Shimura curves – already provide a very rich ground for investigation: geometry, construction of Galois representations from automorphic forms, Jacquet-Langlands correspondence, Diophantine problems (non-emptiness of rational points and Hasse principle),  p-adic uniformization and p-adic geometry, rational and integral models, level-raising and level-lowering problems, etc. The purpose of this lecture series to discuss the construction of integral models of quaternionic Shimura varieties at a good prime and introduce the Ekedahl-Oort stratification.

2. Outline & Descriptions

We shall introduce the notion of canonical models of Shimura varieties due to Shimura, Deligne and others, and discuss briefly the canonical integral models of Shimura varieties modulo a good  prime due to Kisin. The remaining part is to investigate the construction of canonical integral models of quaternion Shimura varieties as well as how to define Newton strata, EO strata and central leaves of the special fiber.  

3. References

[1] P. Deligne, Travaux de Shimura. Seminaire Bourbaki 389 (1970/71), 123--165. LNM 244, 1971.

[2] P. Deligne, Vari'et'es de Shimura: interpr'etation modulaire, et techniques de construction de mod`eles canoniques. Automorphic forms, representations and $L$-functions.

[3] H. Reimann, The semi-simple zeta function of quaternionic Shimura varieties. Lecture Notes in Math., vol. 1657, Springer-Verlag, 1997.

[4] Y. Tian and L. Xiao, On Goren-Oort stratification for quaternionic Shimura varieties, Compos. Math.152 (2016), no. 10, 2134–2220.

4. Registration

https://forms.gle/sQ5D7YcANL6cRJHE8