Integrality properties in the Moduli Space of Elliptic Curves: Isogeny Case

TL;DR

This paper establishes effective bounds on the number of algebraic unit $j$-invariants of elliptic curves, both with fixed $j_0$ and fixed singular modulus, that are isogenous to a given non-CM elliptic curve.

Contribution

It provides the first effective bounds on the count of algebraic units $j$-invariants of isogenous elliptic curves, extending to cases with fixed singular moduli.

Findings

Bounded the number of algebraic unit $j$-invariants for isogenous elliptic curves.

Extended bounds to cases fixing a singular modulus and considering $j - ext{(modulus)}$ as an algebraic unit.

Results are effective, providing explicit bounds for these counts.

Abstract

For a fixed $j$-invariant $j_0$ of an elliptic curve without complex multiplication we bound the number of $j$-invariants $j$ that are algebraic units and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. Our bounds are effective. We also modify the problem slightly by fixing a singular modulus $\alpha$ and looking at all $j$ such that $j-\alpha$ is an algebraic unit and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. The number of such $j$ can again be bounded effectively.