On some arithmetic properties of Siegel functions (II)

TL;DR

This paper explores the construction of normal bases in abelian extensions of imaginary quadratic fields using Siegel functions, providing criteria and methods for specific class field extensions and analyzing their Galois module structures.

Contribution

It introduces new criteria for constructing normal bases of ring class fields and intermediate ray class fields using Siegel functions, extending previous work on Galois module structures.

Findings

Normal bases of ring class fields are constructed using Frobenius determinant relations.

Normal bases of intermediate fields in certain ray class field extensions are found via Kawamoto's arguments.

Galois module structures of specific field extensions are analyzed, extending prior results.

Abstract

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.