Normal bases of ray class fields over imaginary quadratic fields

TL;DR

This paper establishes a criterion for normal bases and proves that certain singular values of Siegel functions form normal bases of ray class fields over most imaginary quadratic fields, advancing the Lang-Schertz conjecture.

Contribution

It introduces a new criterion for normal bases and demonstrates that specific Siegel function singular values generate normal bases in ray class fields over imaginary quadratic fields, excluding only two special cases.

Findings

Singular values of Siegel functions form normal bases over certain imaginary quadratic fields.

The result applies to ray class fields with moduli generated by integers ≥ 2.

Provides an answer to the Lang-Schertz conjecture for these fields.

Abstract

We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer ($\geq2$).