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 demonstrationemmirabilem 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 objectof 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-Weilzeta 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 and1960’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 thatﬁnitely 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 ε-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 LastTheorem.
*The work on this paper was supported by an NSF grant.
ANDREW JOHN WILES
Our approach to the study of elliptic curves is via their associated Galois ¯ representations. Suppose that ρp is the representation of Gal(Q/Q) on the p-division points of an elliptic curve over Q, and suppose for the moment that ρ3 is irreducible. The choice of 3 is critical because a crucial theorem ofLanglands and Tunnell shows that if ρ3 is irreducible then it is also modular. We then proceed by showing that under the hypothesis that ρ3 is semistable at 3, together with some milder restrictions on the ramiﬁcation of ρ3 at the other primes, every suitable lifting of ρ3 is modular. To do this we link the problem, via some novel arguments from commutative algebra, to a class number problem of awell-known type. This we then solve with the help of the paper [TW]. This suﬃces to prove the modularity of E as it is known that E is modular if and only if the associated 3-adic representation is modular. The key development in the proof is a new and surprising link between two strong but distinct traditions in number theory, the relationship between Galois representations and modular forms onthe one hand and the interpretation of special values of L-functions on the other. The former tradition is of course more recent. Following the original results of Eichler and Shimura in the 1950’s and 1960’s the other main theorems were proved by Deligne, Serre and Langlands in the period up to 1980. This included the construction of Galois representations associated to modular forms, the...
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