A subspace theorem approach to integral points on curves
This paper presents a new proof of Siegel's theorem on integral points on affine curves. Siegel's original proof used endomorphisms of the Jacobian; this proof uses, instead, an argument involving the Subspace Theorem of \textit{W. M. Schmidt} [Lecture Notes in Mathematics. 785, Springer (1980; Zbl 0421.10019)]. Specifically, let \(C\) be the affine curve in question. We may assume it is nonsingular. Write it as the complement \(\widetilde C\setminus D\) of a divisor in a smooth projective curve \(\widetilde C\). The proof works by applying Schmidt's Subspace Theorem to the image of \(\widetilde C\) in projective space associated to a complete linear system \(|ND|\), for \(N\) large. This method is similar in spirit to work of \textit{R. Ferretti} [Compos. Math. 121, 247-262 (2000; Zbl 0989.11034)] on integral points on rational varieties. The authors also note that, if the number of points of \(D\) is at least \(3\), then the number of integral points over the ring of integers of a number field \(k\) may be bounded by a number depending only on \(C\) and \([k:\mathbb Q]\), by applying quantitative versions of the Subspace Theorem due to \textit{H. P. Schlickewei} [Compos. Math. 82, 245-273 (1992; Zbl 0751.11033)] and \textit{J.-H. Evertse} [Compos. Math. 101, 225-311 (1996; Zbl 0856.11030)].
scientific article; zbMATH DE number 1744343
scientific article