In June 1993 at the Isaac Newton Institute in Cambridge, England, Andrew Wiles announced in a series of lectures that he had proved Fermat's Last Theorem by proving a large part of the Taniyama-Shimura conjecture on the modularity of elliptic curves. Within a few months it turned out that the announcement was premature and the proof not complete. There had been many such claims before, beginning with Fermat's own 17th-century marginal note, which were never fulfilled. But Wiles persevered, and the last missing step was proved jointly by Richard Taylor and Wiles \ref[Ann. of Math. (2) 141 (1995), no. 3, 553--572; see the following review]. Published less than two years after the original announcement, Wiles' paper contains a proof very close to the original outline he described in his lectures. See the introduction for a very readable step-by-step account of the history of the Wiles and Taylor-Wiles papers. For expository articles about the proof of Fermat's Last Theorem, see papers by F. Q. Gouvêa \ref[Amer. Math. Monthly 101 (1994), no. 3, 203--222; MR 94k:11033], K. A. Ribet \ref[Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 4, 375--402; MR 96b:11073], or K. C. Rubin and A. Silverberg \ref[Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 1, 15--38; MR 94k:11062; erratum; Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 170].

The Taniyama-Shimura conjecture is one of the most important outstanding problems concerning elliptic curves. As formulated by Shimura in the early 1960s, it asserts that every elliptic curve defined over the rational numbers is modular (see below). Wiles proved the modularity of a large class of elliptic curves, including all semistable ones. Although less than the full Taniyama-Shimura conjecture, this result still suffices to prove Fermat's Last Theorem.

Fermat's Last Theorem asserts that there are no positive integers $a$, $b$, $c$, and $n$, with $n > 2$, such that $a^n + b^n = c^n$. To a hypothetical solution of this equation one can associate an elliptic curve $$ E_{a^n,b^n}\colon y^2 = x(x-a^n)(x+b^n). $$ This connection was first noticed and used in the late 1960s by Hellegouarch. In the mid-1980s Frey described how to use this construction to connect Fermat's Last Theorem to the Taniyama-Shimura conjecture. Ribet \ref[Invent. Math. 100 (1990), no. 2, 431--476; MR 91g:11066] was able to prove what Frey suspected: if $a^p + b^p = c^p$ and $p$ is a prime greater than $3$, then $E_{a^pb^p}$ is not modular. In other words, the Taniyama-Shimura conjecture implies Fermat's Last Theorem.

Fix an elliptic curve $E$ and a prime $p$. The Galois group $G_ Q = {\rm Gal}(\overline Q/ Q)$ acts on the group $E(\overline Q)$ of points of $E$ with coordinates in $\overline Q$, and the action on the $p$-power torsion points $E(\overline Q)_{p^\infty} \cong (\bold Q_p/\bold Z_p)^2$ gives rise to a representation $\rho_{E,p} \colon G_ Q \to {\rm GL}_2(\bold Z_p)$. On the other hand, if $f(z) = e^{2 \pi iz} + \sum_{n>1}a_n e^{2\pi inz}$ is a cusp form of weight two for some $\Gamma_0(N) \subset {\rm SL}_2( Z)$, $f$ is an eigenfunction of all Hecke operators, and $\lambda$ is a prime above $p$ of the ring of integers $\scr O_f$ of the number field $ Q(a_2, a_3, \cdots)$, then Eichler and Shimura associated to $f$ a representation $\rho_{f,\lambda} \colon G_ Q \to {\rm GL}_2(\scr O_\lambda)$, where $\scr O_\lambda$ is the completion of $\scr O_f$ at $\lambda$. One says $E$ is modular if $\rho_{E,p}$ "comes from a modular form", i.e., if there exist such a form $f$ and prime ideal $\lambda$ with $\rho_{E,p} \cong \rho_{f,\lambda}$. This definition does not depend on the choice of $p$.

Write $\overline\rho_{E,p}$ and $\overline\rho_{f,\lambda}$ for the representations into ${\rm GL}_2(\overline F_p)$ obtained from $\rho_{E,p}$ and $\rho_{f,\lambda}$ by reduction modulo $p$ and $\lambda$, respectively. Wiles' approach is to prove $E$ is modular in two steps: (1) show that there are $f$ and $\lambda$ such that $\overline\rho_{E,p} \cong \overline\rho_{f,\lambda}$, (2) show that for every representation $\rho \colon G_ Q \to {\rm GL}_2(\scr O_\lambda)$ satisfying certain natural properties, and such that the reduction of $\rho$ modulo $\lambda$ is isomorphic to $\overline\rho_{f,\lambda}$ (so in particular for $\rho = \rho_{E,p}$), there are $f'$ and $\lambda'$ such that $\rho \cong \rho_{f',\lambda'}$.

Although little is known in general about step (1), it is known in the case in which $p=3$ and $\overline\rho_{E,3}$ is irreducible by work of R. P. Langlands \ref[ Base change for ${\rm GL(2)$}, Ann. of Math. Stud., 96, Princeton Univ. Press, Princeton, NJ, 1980; MR 82a:10032] and of J. Tunnell \ref[Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173--175; MR 82j:12015]. (The irreducibility condition turns out not to be a serious problem: when $\overline\rho_{E,3}$ is reducible, Wiles makes very clever use of the irreducible case for $p=3$ to prove step (1) for $p=5$.) The heart of Wiles' proof is his proof of step (2) under certain weak assumptions.

Very briefly, the proof goes as follows. Suppose $E$ is semistable at $p$, $\overline\rho_{E,p}$ is irreducible, and $f$ and $\lambda$ are given by step (1). Write $\scr O$ for $\scr O_\lambda$ and let $k$ be its residue field. B. Mazur's deformation theory of Galois representations \ref[in Galois groups over ${\bf Q$ (Berkeley, CA, 1987)}, 385--437, Springer, New York, 1989; MR 90k:11057], with one important case due to R. Ramakrishna \ref[Compositio Math. 87 (1993), no. 3, 269--286; MR 94h:11054], gives a complete local Noetherian $\scr O$-algebra $R$ with residue field $k$ and a representation $\rho_{R} \colon G_ Q \to {\rm GL}_2(R)$ whose reduction is $\overline\rho_{f,\lambda}$, such that for every complete local Noetherian $\scr O$-algebra $A$ with residue field $k$, every representation $\rho \colon G_ Q \to {\rm GL}_2(A)$ with prescribed ramification properties and whose reduction is $\overline\rho_{f,\lambda}$ factors through ${\rm GL}_2(R)$ by a unique map from $R$ to $A$. In other words, the "universal deformation ring" $R$ parametrizes all "liftings" of $\overline\rho_{f,\lambda}$ with prescribed ramification properties. On the other hand, Wiles constructs a completed Hecke algebra $ T$ which parametrizes all liftings of $\overline\rho_{f,\lambda}$ with the same prescribed ramification properties which come from modular forms. The universal property gives a surjective map $\varphi \colon R \to T$, and to verify step (2) it suffices to show that $\varphi$ is an isomorphism.

Let $\pi \colon T \to {\rm End}(f\scr O) \cong \scr O$ be the map giving the action of $ T$ on the modular form $f$, let $\germ p_ T = \ker(\pi)$, and let ${\germ p_R} = \ker(\pi\circ\varphi)$. Using commutative algebra and algebraic properties of the Hecke algebra $ T$, Wiles shows that the injectivity of $\varphi$ follows from the inequality $\#({\germ p_R}/{\germ p_R}^2) \leq \#(\germ p_ T/\germ p_ T^2)$. (This inequality can be viewed as an analogue of the analytic class number formula for cyclotomic fields, which played a major role in earlier work on Fermat's Last Theorem.) Under certain mild conditions on $\overline\rho_{E,p}$, Wiles reduces the verification of this inequality to the "minimal" case, where the prescribed ramification conditions are as strict as possible. (In this reduction, results and methods of Ribet \ref[in Motives (Seattle, WA, 1991), 639--676, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994; MR 95d:11056] and others developed to attack J.-P. Serre's conjectures \ref[Duke Math. J. 54 (1987), no. 1, 179--230; MR 88g:11022] play a crucial role.) Wiles then proves the desired equality in the minimal case under the assumption that $ T$ is a complete intersection. The verification that $ T$ is a complete intersection is proved in the Taylor-Wiles paper reviewed below.

Reviewed by Karl Rubin