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Letter to J.-F. Mestre. (Lettre à J.-F. Mestre.) (French) Zbl 0629.14016

Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 263-268 (1987).
[For the entire collection see Zbl 0615.00004.]
This paper is the author’s now famous letter to Mestre which detailed several conjectures concerning Galois representations and modular forms. In large part the importance of some of these conjectures lies in the fact that they provide justification for Frey’s beautiful idea that Fermat’s last theorem should follow from the Taniyama-Weil conjecture that all elliptic curves over \({\mathbb{Q}}\) are modular.
Shortly after the author made these conjectures, Ribet (see [K. A. Ribet and W. A. Stein, in: Arithmetic algebraic geometry. Expanded lectures delivered at the graduate summer school of the Institute for Advanced Study/Park City Mathematics Institute, Park City, UT, USA, June 20–July 10, 1999. Providence, RI: American Mathematical Society (AMS). 143–232 (2001; Zbl 1160.11327)]) was able to prove a version that was sufficiently strong to show that Fermat’s last theorem does indeed follow from Weil-Taniyama. Ribet’s proof combined techniques of Mazur with a study of the interplay between the jacobians of certain Shimura curves and the jacobians of certain classical modular curves.
See also the author’s paper [Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)].
Reviewer: S.Kamienny

MSC:

14G25 Global ground fields in algebraic geometry
11R32 Galois theory
14H25 Arithmetic ground fields for curves
11F12 Automorphic forms, one variable