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Rational elliptic curves are modular. (English) Zbl 0998.11030
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque 276, 161-188, Exp. No. 871 (2002).
After the modularity of any semistable elliptic curve $$E$$ over the rationals had been proved by Wiles (and Taylor), it still was desirable to get rid of the semistability assumption. Step by step the class of elliptic curves known to be modular was enlarged, a program which was finally finished in [C. Breuil, B. Conrad, F. Diamond, and R. Taylor, J. Am. Math. Soc. 14, 843-939 (2001; Zbl 0982.11033)].
In the paper under review, the author describes Wiles’ original strategy of proof and then outlines what had to be improved and how this could be done in order to get the full Shimura-Taniyama conjecture. He ends up by stating some more recent results on the modularity of Galois-representations.
For the entire collection see [Zbl 0981.00011].

##### MSC:
 11G05 Elliptic curves over global fields 11F80 Galois representations 11F03 Modular and automorphic functions
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