On Siegel invariants of certain CM-fields

TL;DR

This paper constructs Siegel invariants for certain CM-fields using theta constants, generalizing known invariants for imaginary quadratic fields, and demonstrates their role in generating ray class fields through Galois actions and numerical examples.

Contribution

It introduces a new construction of Siegel invariants for CM-fields and explores their Galois properties, extending classical invariants to a broader class of fields.

Findings

Siegel invariants are expressed via special values of theta constants.

Galois actions on these invariants are explicitly described.

Numerical examples confirm the invariants generate the ray class field.

Abstract

We first construct Siegel invariants of some CM-fields in terms of special values of theta constants, which would be a generalization of Siegel-Ramachandra invariants of imaginary quadratic fields. And, we further describe Galois actions on these invariants and provide some numerical examples to show that this invariant really generates the ray class field of a CM-field.