Runge's Method and Modular Curves

Details Runge's Method and Modular Curves Runge's Method and Modular Curves

2009-07-19

arxiv.org/abs/0907.3306v1

Mathematics

Number Theory

Nineteen pages

Scientific paper

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.

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