  We bound the j-invariant of S-integral points on arbitrary modular curves
over arbitrary fields, in terms of the congruence group defining the curve,
assuming a certain Runge condition is satisfied by our objects. We then apply
our bounds to prove that for sufficiently large prime p, the points of $X_0^+
(p^r)(Q)$ with r>1 are either cusps or CM points. This can be interpreted as
the non-existence of quadratic elliptic Q-curves with higher prime-power
degree.
