Cambridge Journal of Mathematics

Volume 9 (2021)

Number 1

On the structure of some $p$-adic period domains

Pages: 213 – 267



Miaofen Chen (School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China)

Laurent Fargues (CNRS, Institut de Mathématiques de Jussieu, Paris, France)

Xu Shen (Morningside Center of Mathematics, Academy of Mathematics and Systems Science, C.A.S., Beijing, China)


We study the geometry of the $p$‑adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues–Rapoport conjecture for $p$‑adic period domains: for a reductive group $G$ over a $p$‑adic field and a minuscule cocharacter $\mu$ of $G$, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set $B(G, \mu)$ is fully Hodge–Newton decomposable.

2010 Mathematics Subject Classification

Primary 11G18. Secondary 14G20.

Miaofen Chen was partially supported by NSFC grant No. 11671136, No. 12071135 and STCSM grant No. 18dz2271000.

Laurent Fargues was partially supported by ANR grant ANR-14-CE25-0002 “PerCoLaTor” and ERC Advanced grant 742608 “GeoLocLang”.

Xu Shen was partially supported by the Chinese Academy of Sciences grants 50Y64198900, 29Y64200900, the Recruitment Program of Global Experts of China, the NSFC grants No. 11631009 and No. 11688101, and the National Key R&D Program of China 2020YFA0712600.

Received 20 March 2019

Published 27 October 2021