文档介绍：Annals of Mathematics, 141 (1995), 443-552
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 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 ∼ 1637
Abstract. When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The
Last Problem and was so impressed byit 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 corollarybyvirtue of work byFrey,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