# Integrality properties in the Moduli Space of Elliptic Curves: CM Case

## TL;DR

This paper proves finiteness results for singular moduli related to algebraic units, providing explicit bounds where previous work only established finiteness non-effectively.

## Contribution

It extends Habegger's finiteness result by giving explicit bounds for singular moduli where the difference with a fixed algebraic number is an algebraic unit.

## Key findings

- Finiteness of singular moduli with algebraic unit differences established.
- Explicit bounds provided for the number of such singular moduli.
- Generalization to cases with fixed non-CM elliptic curve invariants.

## Abstract

A result of Habegger shows that there are only finitely many singular moduli such that $j$ or $j-\alpha$ is an algebraic unit. The result uses Duke's Equidistribution Theorem and is thus not effective. For a fixed $j$-invariant $\alpha \in \bar{\mathbb{Q}}$ of an elliptic curve without complex multiplication, we prove that there are only finitely many singular moduli $j$ such that $j-\alpha$ is an algebraic unit. The difference to the work of Habegger is that we give explicit bounds.