The Action of Complex Conjugation on a Shimura Variety

Abstract

In the case that a Shimura variety has a real canonical model, complex conjugation defines an involution of the set of complex points of the variety. It is necessary to have an explicit description of this involution in order, for example, to compute the zeta function of the variety. Langlands conjectured such a description in [1, p. 417-18] and our purpose is to prove his conjecture for all Shimura varieties of abelian type. (This class of Shimura varieties is defined in S 1 of this paper; it contains all those whose canonical model is known, at the time of writing, to exist.) Recall that a Shimura variety Sh(G, X) is defined by a Q-rational reductive group G and a family X of homomorphisms COx -> G(R) satisfying certain conditions. Initially Sh (G, X) is defined as a complex variety but is expected to have a model over a certain number field E(G, X) called the reflex field. A canonical model for Sh(G, X) is a variety M(G, X) over E(G, X) satisfying certain conditions sufficient to determine it uniquely. Assume that E(G, X) is real and that the canonical model exists so that complex conjugation defines an involution 0 of Sh(G, X). When the canonical model is a moduli variety, and so has a direct description, the proof of the conjecture is straightforward. This is not usually the case, and just as the construction of the canonical model is intricate in general, so must be the proof of the conjecture. In particular, it must involve an analogous assertion for connected Shimura varieties. Such an assertion is proved in Shih [1] for connected Shimura varieties that are of primitive abelian type C, and this result is the starting point of our proof of the conjecture for all Shimura varieties of abelian type. (Since a does not preserve the connected component the analogous assertion takes on quite a different form from the original; it becomes rather a statement about the action of a negative element of G(Q).) To see how our result relates to the zeta function, consider an arbitrary