On the method of Coleman and Chabauty
\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.
cardinality of rational points on a curve
scientific article; zbMATH DE number 683114
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