Shimura varieties and complex conjugation

Work of Shimura, Langlands, Milne-Shih, and, more recently,
Taylor, have examined the action of complex conjugation on Shimura
varieties. We study this topic from the viewpoint of the general theory of
these varieties. In this talk, we will concentrate on certain Shimura
varieties whose reflex field E is a CM field and stablish descent of these
to the maximal totally real subfield E^+. We will discuss a strategy to
construct these descent data in more generality, and mention possible
applications. This work is joint with Don Blasius (UCLA) and remains in
progress.