Special values of Hilbert modular functions (Q1097915)

In a recent series of papers by W. L. Baily on special values of arithmetic Hilbert modular functions, the order of complex multiplications associated to a ``special point'' or ``CM-point'' was required to be the maximal order in its quotient field and one of the objects of this paper is to remove this restriction of maximality. The author's main result describes how Galois automorphisms move ``CM- points''; an immediate consequence is that the value of any arithmetic Hilbert modular function at any such point is abelian over the associated ``reflex field''. The author also confirms a result of Tate on the effect of arbitrary Galois automorphisms of the algebraic closure of \({\mathbb{Q}}\) on ``CM-points''.