Isolated points on modular curves of prime-power level

Abstract

An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by $\mathbf{P}^1$ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of $j$-invariants, in extensions of bounded degree, that arise as the $j$-invariant of an isolated point on a modular curve. We obtain finiteness results on these sets for families of modular curves with prime-power level. This is related to recent work of Bourdon and Ejder, who classified rational $j$-invariants of isolated points on the families $X_1(n)$ and $X_0(n)$, for $n$ a prime power.