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**Number theory as gadfly.**
*(English)*
Zbl 0764.11021

In this article the author seeks to provide his audience with a basic understanding of a statement of the Shimura-Taniyama-Weil conjecture. He concentrates on wending his way through various difficult notions, giving as simple and straightforward an explanation as possible, while still providing some motivation. He goes on to sketch the proof of the Frey- Serre-Ribet result that this conjecture implies the truth of Fermat’s Last Theorem. For anyone unfamiliar with this area (including the reviewer) this paper provides excellent inspiration to learn more about the subject.

The author also makes various intriguing statements and speculations, for instance, that progress with the Shimura-Taniyama-Weil conjecture might come from any of many different directions, such as the study of differential geometry, or partial differential equations, or …representation theory of reductive groups, or …Kac-Moody algebras, or …‘ideas that have been, or will be, imported from Physics…’!!

The author also makes various intriguing statements and speculations, for instance, that progress with the Shimura-Taniyama-Weil conjecture might come from any of many different directions, such as the study of differential geometry, or partial differential equations, or …representation theory of reductive groups, or …Kac-Moody algebras, or …‘ideas that have been, or will be, imported from Physics…’!!

Reviewer: A.Granville (Athens / Georgia)

### MSC:

11D41 | Higher degree equations; Fermat’s equation |

11G05 | Elliptic curves over global fields |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |