Isolated Points on $X_1(\ell^n)$ with rational $j$-invariant}

TL;DR

This paper classifies isolated points with rational j-invariants on certain modular curves, showing such points are extremely rare and explicitly identifying the only possible cases for prime  and specific j-invariants.

Contribution

The paper proves that non-CM isolated points with rational j-invariants on X_1(^n) occur only for =37 with specific j-invariants, providing a precise classification.

Findings

Identifies =37 as the unique prime for such points.

Specifies the only possible rational j-invariants for these points.

Shows the reverse implication holds for one j-invariant but remains open for the other.

Abstract

Let $\ell$ be a prime and let $n\geq 1$. In this note we show that if there is a non-cuspidal, non-CM isolated point $x$ with a rational $j$-invariant on the modular curve $X_1(\ell^n)$, then $\ell=37$ and the $j$-invariant of $x$ is either $7\cdot11^3$ or $-7.137^3\cdot2083^3$. The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.