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On Goren–Oort stratification for quaternionic Shimura varieties

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  21 September 2016

Yichao Tian  and
Liang Xiao
Yichao Tian
Affiliation:
Morningside Center of Mathematics, Chinese Academy of Sciences, 55 Zhong Guan Cun East Road, Beijing 100190, China email yichaot@math.ac.cn
Liang Xiao
Affiliation:
UConn Department of Mathematics, 196 Auditorium Road, Unit 3009, Storrs, CT 06269-3009, USA email liang.xiao@uconn.edu
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Abstract

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Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic-$p$ fiber of a quaternionic Shimura variety of maximal level at $p$. We show that each stratum is a $(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer $r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$.

Keywords

quaternionic Shimura varietiesGoren–Oort stratificationp-divisible groups

MSC classification

Primary: 14G35: Modular and Shimura varieties 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 10 , October 2016 , pp. 2134 - 2220
Copyright
© The Authors 2016 

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