Runge's Method and Modular Curves
Yuri Bilu, Pierre Parent · International Mathematics Research Notices · 2010
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 with r < 1 are either cusps or complex multiplication points. This can be interpreted as the non-existence of quadratic elliptic -curves with higher prime-power degree.