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Fermat’s Last Theorem: From Fermat to Wiles. (English) Zbl 0849.11004

This expository paper originates from the author’s forthcoming book (of the same name as this note). It presents a fairly detailed discussion of various notions underlying Wiles’ proof. After a very brief introduction the author discusses several areas of number theory on which Wiles’ work impacts, particularly the Birch-Swinnerton-Dyer conjecture and the Herbrand-Ribet theorem (construction of unramified extensions). The third part of this paper is a rather solid outline dealing with technical aspects of Wiles’ proof of Fermat’s Last Theorem. The survey concludes with a useful timetable of the announcements pertaining to Wiles’ proof.
Both the well-tutored student, looking for preparation for detailed study of the technicalities of the proof, and the reader looking for insight into the underlying philosophies, will find much interest in the author’s remarks. In contrast, incidentally, the reviewer’s book ‘Notes on Fermat’s Last Theorem’ [Wiley-Interscience (1996)] is not even quite an introduction to the ideas required to benefit properly from the present survey.

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11G05 Elliptic curves over global fields
11D41 Higher degree equations; Fermat’s equation
11F11 Holomorphic modular forms of integral weight
11-03 History of number theory
14H52 Elliptic curves
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