Isolated points on modular curves of prime-power level

An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by P¹ 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 X₁(n) and X₀(n), for n a prime power.