Runge's Method and Modular Curves

TL;DR

This paper establishes bounds on the j-invariant of S-integral points on modular curves using Runge's method, and applies these bounds to show certain points are only cusps or CM points, implying non-existence results for quadratic elliptic curves with high prime-power degree.

Contribution

It introduces a new application of Runge's method to bound j-invariants on modular curves and derives non-existence results for specific elliptic curves over Q.

Findings

Bound on j-invariant of S-integral points in terms of congruence group

For large primes p, points on X_0^+(p^r) are only cusps or CM points

Non-existence of quadratic elliptic Q-curves with high prime-power degree

Abstract

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.