Remark on a characterization of certain ring class fields by their absolute Galois group

Abstract

where A(co) stands for the discriminant of the complex lattice generated by 1 and to and g2(u) is the Weierstrass invariant of this lattice. If ti=Q(\/ — D) is an imaginary quadratic number field, it is well known from the theory of class fields with complex multiplication that the singular values j(a), where a£u, la >0, generate algebraic number fields which are abelian over fi, namely the so-called ring class fields. Detailed references for the literature on the theory of ring class fields may be found in the report of Deuring [l]. In the rather extensive theory of ring class fields it is shown that ®Uia))/Q iS a normal extension with its Galois group © being an extension of the abelian Galois group s of Q(j(a))/Cl, completely determined by the relations