# On singular moduli that are S-units

## TL;DR

This paper investigates conditions under which singular moduli can be S-units, proving that if S contains only primes congruent to 1 modulo 3, then no singular modulus is an S-unit, extending previous results about units.

## Contribution

The paper extends known results by identifying specific prime sets S for which singular moduli cannot be S-units, and analyzes norm factorizations of certain singular moduli.

## Key findings

- No singular modulus is an S-unit when S contains only primes ≡ 1 mod 3
- Provides remarks on the general case of S-units for singular moduli
- Studies norm factorizations of a special family of singular moduli

## Abstract

Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.