On the Zeta-Functions of Some Simple Shimura Varieties

In an earlier paper [ 14 ] I have adumbrated a method for establishing that the zeta-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [ 13 ]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [ 13 ] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [ 17 ].