Fields generated by sums and products of singular moduli

TL;DR

This paper investigates the fields generated by sums and products of singular moduli, revealing that such fields are typically generated by these sums or products unless the moduli are conjugate, in which case the generated field is of small degree.

Contribution

The paper establishes new results on the generators of fields formed by singular moduli, showing that sums and products usually generate the entire field unless the moduli are conjugate.

Findings

The field generated by two singular moduli is usually generated by their sum.

Similarly, the field generated by their product is often generated by the product.

If the singular moduli are conjugate, the generated field has degree at most 2.

Abstract

We show that the field $\mathbb{Q}(x,y)$, generated by two singular moduli~$x$ and~$y$, is generated by their sum ${x+y}$, unless~$x$ and~$y$ are conjugate over~$\mathbb{Q}$, in which case ${x+y}$ generates a subfield of degree at most~$2$. We obtain a similar result for the product of two singular moduli.