Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$

TL;DR

This paper classifies rational $j$-invariants arising from isolated points on modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$, completing previous partial classifications and providing a finite list of such invariants.

Contribution

The paper fully classifies rational $j$-invariants from isolated points on $X_1(\ell^n)$ and $X_0(\ell^n)$, extending prior partial results.

Findings

15 rational $j$-invariants from isolated points on $X_1(\ell^n)$

19 rational $j$-invariants from isolated points on $X_0(\ell^n)$

Complete classification of isolated rational $j$-invariants on these modular curves

Abstract

Let $\ell$ and $n$ be positive integers with $\ell$ prime. The modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$ are algebraic curves over $\mathbb{Q}$ whose non-cuspidal points parameterize elliptic curves with a distinguished point of order $\ell^n$ or a distinguished cyclic subgroup of order $\ell^n$, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 $j$-invariants in $\mathbb{Q}$ which arise as the image of an isolated point $x\in X_1(\ell^n)$ under the natural map $j:X_1(\ell^n) \rightarrow X_1(1)$. This completes a prior partial classification of Ejder. We also identify the 19 rational $j$-invariants which correspond to isolated points on $X_0(\ell^n)$.