On some arithmetic properties of Siegel functions (II)

Abstract

Let K be an imaginary quadratic field of discriminant d(K) <= -7. We deal with problems of constructing normal bases between abelian extensions of K by making use of singular values of Siegel functions. First, we find normal bases of ring class fields of orders of bounded conductors depending on d(K) over K by using a criterion deduced from the Frobenius determinant relation. Next, denoting by K-(N) the ray class field modulo N of K for an integer N >= 2 we consider the field extension K-(p(m))2/K-(pm) for a prime p >= 5 and a positive integer m relatively prime to p and then find normal bases of all intermediate fields over K-(pm) by utilizing Kawamoto's arguments. We further investigate certain Galois module structure of the field extension K-(p(m))n/K-(p(m))l with n >= 2l, which would be an extension of Komatsu's work.