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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.

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