# Úttimo teorema de fermat

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Annals of Mathematics, 141 (1995), 443-551

Modular elliptic curves and Fermat’s Last Theorem
By Andrew John Wiles* For Nada, Claire, Kate and Olivia
Pierre de Fermat Andrew John Wiles

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, 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 ∼ 1637
Abstract. 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 . The 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 previous work by Frey, Serre and Ribet.

Introduction An 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 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