A subspace theorem approach to integral points on curves
type: Item
instance of: scholarly article
title: A subspace theorem approach to integral points on curves
zbMATH Open document ID: 1012.11051
DOI: 10.1016/S1631-073X(02)02240-9
author: Pietro Corvaja
author: Umberto Zannier
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-07-24T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 1744343
zbMATH Keywords: Siegel's theorem
zbMATH Keywords: integral point on affine curves
zbMATH Keywords: Subspace Theorem
reviewed by: Paul Vojta
MaRDI profile type: Publication
Wikidata QID: Q56699706
cites work: Q4884650
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Q3931657
cites work: Q4486426
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cites work: Diophantine approximation
cites work: Diophantine approximations and diophantine equations
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cites work: Diophantine approximations and value distribution theory
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description: scientific article; zbMATH DE number 1744343