On the Field of Definition for a Field of Automorphic Functions: II

Abstract

The purpose of this paper is to complement the main theorems of the previous papers with the same title, referred to hereafter as [FI] and [FII], and to simplify the proofs. In [FI, II], it was shown that a certain abelian extension k of a certain algebraic number field K' can be determined by a family 1(&2) of abelian varieties belonging to a given type 12 of structures. The structures consist of endomorphism algebra, polarization and points of finite order, of an abelian variety. The case of structures without points of finite order was treated in [FI]. We observed that k is unramified over K', and the law of reciprocity for the extension k/K' can be described in terms of the interrelation between 12 and its transform. One of the aims of [FII] was to extend this result to the general case involving points of finite order and ramified abelian extensions. However, we determined only the ideal group in K' corresponding to k, but did not give an explicit reciprocity law in such a general case. This will be done in the present paper. We shall also reformulate the results in a more comprehensible manner, and give a simplified proof by means of the characterization of the field k provided by [2, 5.1]. This characterization enables us to dispense with artificial considerations which were necessary in the original proofs. We devote one section to clarify a point concerning the existence of a special member of the family Y(f2), of which the previous proof in [FI] was not with complete clarity. List of symbols. They were introduced in [FI, II], occasionally with a somewhat general meaning.