On the method of Coleman and Chabauty
type: Item
instance of: scholarly article
title: On the method of Coleman and Chabauty
zbMATH Open document ID: 0824.14017
DOI: 10.1007/BF01459799
author: William G. McCallum
published in: Mathematische Annalen
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165223
review text: \textit{R. F. Coleman} [``Effective Chabauty'', Duke Math. J. 52, 765-770 (1985; Zbl 0588.14015)], following Chabauty, has shown how to obtain good bounds on the cardinality of \(K\)-rational points \((K\) number field) on a curve of genus \(g \geq 2\), if the rank of the Mordell-Weil group \(J(K)\) of the corresponding Jacobian \(J\) is less than \(g\). -- The author applies these methods to the Fermat curve \(x^ p + y^ p = 1\). And he shows under certain assumptions that the second case of Fermat's Last Theorem is true. In the first part it is shown that the conditions on the rank of \(J(K)\) in the Coleman-Chabauty paper (loc. cit.) are satisfied. Here the author has to assume the existence of suitable cocycles in the Selmer group. -- Then the second part contains a detailed applications of Coleman's method to the Fermat curve and other related curves. In the case that the prime number \(p\) is regular the existence of these cocycles can be guaranteed. This provides an independent proof of Kummer's result, also independent of the recent work of \textit{A. Wiles} [cf. Ann. Math., II. Ser. 141, No. 3, 443-551 (1995; Zbl 0823.11029)]. The paper shows in a good way how Coleman's method can be fashioned into a quite precise tool for bounding rational points on curves.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 683114
zbMATH Keywords: effective Chabauty
zbMATH Keywords: cardinality of rational points on a curve
zbMATH Keywords: Fermat's Last Theorem
zbMATH Keywords: rank of the Mordell-Weil group
zbMATH Keywords: Fermat curve
zbMATH Keywords: rational points on curves
MaRDI profile type: Publication
cites work: The first case of Fermat's last theorem
cites work: Q4081425
cites work: Q5532660
cites work: Torsion points on curves and p-adic abelian integrals
cites work: Effective Chabauty
cites work: Q5332674
cites work: Q3931542
cites work: Some results on the Mordell-Weil group of the Jacobian of the Fermat curve
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve
cites work: The arithmetic of Fermat curves
cites work: Q3754095
cites work: Spécialisation du foncteur de Picard
cites work: Q3904661
cites work: Q3277020
cites work: Q3943047
OpenAlex ID: W2028959974
description: scientific article; zbMATH DE number 683114