Complex Conjugation and Descent of Shimura Varieties

Shimura varieties are defined over a canonically defined number field called the reflex field.
It is either totally real or a totally imaginary quadratic extension of a totally real field, i.e. a CM
field.  In this talk we consider only the latter case, and show that in many cases there exists a
descent of the  variety to the maximal totally real subfield of the reflex field. By choice, we
use only of the defining group theoretic data of the variety as well as general properties of
Shimura varieties, algebraic groups,etc.  In other words, no reference to moduli problems
is needed. The existence of these descents suggests several further problems. (This is joint
work with Lucio Guerberoff)