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

Abstract

Let $\ell$ be a prime and let $n\ge 1$. 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 $7\cdot 11^3$ or $-7.137^3\cdot 2083^3$. The reverse implication holds for the first j-invariant but it is currently unknown whether or not it holds for the second.

6 pages, accepted to Research in Number Theory