An explicit bound for integral points on modular curves (Q6580580)

Let \(X\) be a smooth connected projective algebraic curve defined over a number field \(K\) and let \(x \in K(X)\) be a non-constant rational function on \(X\). In the case where \(X=X_{\Gamma}\) is the modular curve corresponding to a congruence subgroup \(\Gamma\) of \(\text{SL}_2(\mathbb{Z})\) and \(x=j\) is the \(j\)-invariant, let \(h\) be the standard absolute logarithmic height defined on the set \(\overline{\mathbb{Q}}\) of algebraic numbers. There is a bound depending on a constant \(C\) for these heights \(h\) of the integral points on modular curves.\N\NIn the present paper, the author determines this constant \(C\) by using an explicit Baker's inequality to obtain an explicit bound.