On some theta constants and class fields

TL;DR

This paper establishes conditions for products of theta constants to be modular functions and describes subfields of ray class fields generated by special theta constant values using Shimura's reciprocity law.

Contribution

It provides a new sufficient condition for theta constant products to be modular functions and explicitly describes subfields of ray class fields generated by theta constants.

Findings

Derived a sufficient condition for theta constants to be modular functions

Described subfields of ray class fields generated by theta constants

Applied Shimura's reciprocity law to explicit class field generation

Abstract

We first find a sufficient condition for a product of theta constants to be a Siegel modular function of a given even level. And, when $K_{(2p)}$ denotes the ray class field of $K=\mathbb{Q}(e^{2\pi i/5})$ modulo $2p$ for an odd prime $p$, we describe a subfield of $K_{(2p)}$ generated by the special value of certain theta constant by using Shimura's reciprocity law.