Runge's Method and Modular Curves

We bound the j-invariant of S-integral points on arbitrary modular curves over arbitrary number 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 complex multiplication points. This can be interpreted as the non-existence of quadratic elliptic Q-curves with higher prime-power degree.