Complex conjugation and Shimura varieties

TL;DR

This paper investigates how complex conjugation acts on Shimura varieties and demonstrates conditions under which these varieties can be descended to their maximal totally real fields, focusing on cases with specific group types.

Contribution

It proves the existence of descent for many Shimura varieties with certain adjoint group factors, expanding understanding of their field of definition.

Findings

Descent exists for Shimura varieties with adjoint groups of type A or D

Includes a large family of Shimura varieties of abelian type

Constructs are carried out at the level of Shimura data and group theory

Abstract

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending these to the maximal totally real field of the reflex field. We prove the existence of such descent for many Shimura varieties whose associated adjoint group has certain factors of type A or D. This includes a large family of Shimura varieties of abelian type. Our considerations and constructions are carried out purely at the level of Shimura data and group theory.