On some arithmetic properties of Siegel functions (II)

Let $K$ be an imaginary quadratic field with discriminant $d_K\leq-7$. We deal with problems of constructing normal bases between abelian extensions of $K$ by making use of singular values of Siegel functions. First, we show that a criterion achieved from the Frobenius determinant relation enables us to find normal bases of ring class fields of orders of bounded conductors depending on $d_K$ over $K$. Next, denoting by $K_{(N)}$ the ray class field modulo $N$ of $K$ for an integer $N\geq2$ we consider the field extension $K_{(p^2m)}/K_{(pm)}$ for a prime $p\geq5$ and an integer $m\geq1$ relatively prime to $p$ and then find normal bases of all intermediate fields over $K_{(pm)}$ by utilizing Kawamoto's arguments. And, we further investigate certain Galois module structure of the field extension $K_{(p^{n}m)}/K_{(p^{\ell}m)}$ with $n\geq 2\ell$, which would be an extension of Komatsu's work.