Improved bounds for integral points on modular curves using Runge's method

Abstract

Consider a modular curve $X_G$ defined over a number field $K$, where $G$ is a subgroup of $G{L}_2(\mathbb{Z}/N\mathbb{Z})$ with $N>2$. The curve $X_G$ comes with a morphism $j:X_G\to {\mathbb{P}}_K^1={\mathbb{A}}_K^1\cup \{\infty \}$ to the $j$-line. For a finite set of places $S$ of $K$ that satisfies a certain condition, Runge's method shows that there are only finitely many points $P\in X_G(K)$ for which $j(P)$ lies in the ring ${\mathfrak{O}}_{K,S}$ of $S$-units of $K$. We prove an explicit version which shows that if $j(P)\in {\mathfrak{O}}_{K,S}$ for some $P\in X_G(K)$, then the absolute logarithmic height of $j(P)$ is bounded above by $N^{12}\log N$. Explicits upper bounds have already been obtained by Bilu and Parent though they are not polynomial in $N$. The modular functions needed to apply Runge's method are constructing using Eisenstein series of weight $1$.