Complex conjugation and Shimura varieties

Abstract

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending Shimura varieties to the maximal totally real field of the reflex field. We prove the existence of such a 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.