Annals of Mathematics,142(1995),443-551 Modular elliptic curves and Fermat's Last Theorem By ANDREW WILES* For Nada,Clare,Kate and Olivia Cubum autem in duos cubos,aut quadratoguadratum in duos m in infinitum cujus ret demonstrationem mirabilem sane deteri.Hanc marginis exiguitas non caperet. Pierre de fermat Introduction An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form Xo(N).Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a ctional equation of the standard type.If an elliptic curve over Q with a given j-invariant is modular it ise ves 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 of Weil in 1967 Wel (as an xercise for the interested r der!),in which. moreover,Weil gave co eptual evidence for the conjecture.Alth ugh it hac been numerically verified in many cases,prior to the results described in this paper it had only been known that finitely many j-invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should PeratsLast Theorem.The precise mechanism relting the err as the e-conj ctur e and this was then proved by Ribet in the summer of 1986.Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's Last Theorem. The workon this paper was supported byan NSF grant.Annals of Mathematics, 142 (1995), 443-551 Modular elliptic curves and Fermat 's Last Theorem For Nada, Clare, Kate and Olivia Cubum autem in duos cubos, aut quadratoquadratum in duos quadra￾toquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. Pierre de Femnat Introduction An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form Xo(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 of Weil in 1967 [We] (as an exercise for the interested reader!), in which, moreover, Weil gave conceptual evidence for the conjecture. Although it had been numerically verified in many cases, prior to the results described in this paper it had only been known that finitely many j-invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should imply Fermat's Last Theorem. The precise mechanism relating the two was formulated by Serre as the &-conjecture and this was then proved by Ribet in the summer of 1986. Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's Last Theorem. *The work on this paper was supported by an NSF grant