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Annals of Mathematics, 141 (1995), 443-552Modular elliptic curves andFermat’s Last TheoremBy Andrew John Wiles*For Nada, Claire, Kate and OliviaPierre de FermatAndrew John WilesCubum autem in duos cubos, aut quadratoquadratum in duos quadra- toquadratos, et generaliter nullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere: cujes rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.- Pierre de Fermat ∼ 1637Abstract. When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. This object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of work by Frey, Serre and Ribet.IntroductionAn elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form X0(N ). Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a functional equation of the standard type. If an elliptic curve over Q with a given j-invariant is modular then it is easy to see that all elliptic curves with the same j-invariant are modular (in which case we say that the j-invariant is modular). A well-known conjecture which grew out of the work of Shimura and Taniyama in the 1950’s and 1960’s asserts that every elliptic curve over Q is modular. However, it only became widely known through its publication in a paper o...