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2021 | OriginalPaper | Buchkapitel

The S-Integral Points on the Projective Line Minus Three Points via Finite Covers and Skolem’s Method

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Abstract

We describe a p-adic proof of the finiteness of \((\mathbb {P}^1-\{0,1,\infty \})(\mathbb {Z}[S^{-1}])\) using only Skolem’s method applied to finite covers.
Literatur
[Bou98]
Zurück zum Zitat Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation. MR1728312 (2001g:17006) Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation. MR1728312 (2001g:17006)
[Cha41]
Zurück zum Zitat Claude Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885 (French). MR0004484 (3,14d) Claude Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885 (French). MR0004484 (3,14d)
[Kim05]
Zurück zum Zitat Minhyong Kim, The motivic fundamental group of P 1∖{0,  1, ∞} and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. MR2181717 (2006k:11119) Minhyong Kim, The motivic fundamental group of P 1∖{0,  1, } and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. MR2181717 (2006k:11119)
[MP12]
Zurück zum Zitat William McCallum and Bjorn Poonen, The method of Chabauty and Coleman, Explicit Methods in Number Theory: Rational Points and Diophantine Equations, Panoramas et Synthèses, vol. 36, Société Mathématique de France, Paris, 2012, pp. 99–117. MATH William McCallum and Bjorn Poonen, The method of Chabauty and Coleman, Explicit Methods in Number Theory: Rational Points and Diophantine Equations, Panoramas et Synthèses, vol. 36, Société Mathématique de France, Paris, 2012, pp. 99–117. MATH
[Sie26]
Zurück zum Zitat Carl Ludwig Siegel, The integer solutions of the equation y 2 =  ax n +  bx n−1 + … +  k, J. London Math. Soc. 1 (1926), 66–68. MathSciNet Carl Ludwig Siegel, The integer solutions of the equation y 2 =  ax n +  bx n−1 + … +  k, J. London Math. Soc. 1 (1926), 66–68. MathSciNet
[Sko34]
Zurück zum Zitat Th. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, 8. Skand. Mat.-Kongr., Stockholm, 1934, pp. 163–188 (German). Th. Skolem, Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und diophantischer Gleichungen, 8. Skand. Mat.-Kongr., Stockholm, 1934, pp. 163–188 (German).
[Tri19]
Zurück zum Zitat Nicholas George Triantafillou, Restriction of scalars, the Chabauty-Coleman method, and \(\mathbb {P}^1 \backslash \{0, 1, \infty \}\), June 2019. Ph.D. thesis, Massachusetts Institute of Technology. Nicholas George Triantafillou, Restriction of scalars, the Chabauty-Coleman method, and \(\mathbb {P}^1 \backslash \{0, 1, \infty \}\), June 2019. Ph.D. thesis, Massachusetts Institute of Technology.
Metadaten
Titel
The S-Integral Points on the Projective Line Minus Three Points via Finite Covers and Skolem’s Method
verfasst von
Bjorn Poonen
Copyright-Jahr
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_21

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