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Runge’s method and modular curves. (English) Zbl 1304.11054
Summary: 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 \(X_0^+ (p^r) (\mathbb Q)\) with \(r < 1\) are either cusps or complex multiplication points. This can be interpreted as the non-existence of quadratic elliptic \(\mathbb Q\)-curves with higher prime-power degree.

11G18 Arithmetic aspects of modular and Shimura varieties
11G16 Elliptic and modular units
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