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 Fermat’s Last Theorem states that the equation xn + yn = zn, xyz 6= 0 has no integer solutions when n is greater than or equal to 3. Around 1630, Pierre de Fermat claimed that he had found a “truly wonderful” proof of this theorem, but that the margin of his copy of Diophantus’ Arithmetica was too small to contain it: “Cubum autem in duos cubos, aut quadrato quadratum in duos quadrato quadratos, et generaliter nullam in inﬁnitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere; cujus rei demonstrationem mirabile sane detexi. Hanc marginis exiguitas non caperet.” Among the many challenges that Fermat left for posterity, this was to prove the most vexing. A tantalizingly simple problem about whole numbers, it stood unsolved for more than 350 years, until in 1994 Andrew Wiles ﬁnally laid it to rest. Prehistory: The only case of Fermat’s Last Theorem for which Fermat actually wrote down a proof is for the case n = 4. To do this, Fermat introduced the idea of inﬁnite descent which is still one the main tools in the study of Diophantine equations, and was to play a central role in the proof of Fermat’s Last Theorem 350 years later. To prove his Last Theorem for exponent 4, Fermat showed something slightly stronger, namely that the equation x4+y4 = z2 has no solutions in relatively prime integers with xyz 6= 0. Solutions to such an equation correspond to rational points on the elliptic curve v2 = u3 −4u. Since every integer n ≥ 3 is divisible either by an odd prime or by 4, the result of Fermat allowed one to reduce the study of Fermat’s equation to the case where n = ` is an odd prime. In 1753, Leonhard Euler wrote down a proof of Fermat’s Last Theorem for the exponent ` = 3, by performing what in modern language we would call a 3-descent on the curve x3 + y3 = 1 which is also an elliptic curve. Euler’s argument (which seems to have contained a gap) is explained in [Edw], ch. 2, and [Dic1], p. 545. It took mathematicians almost 100 years after Euler’s achievement to handle the case ` = 5; this was settled, more or less simultaneously, by Gustav 3 Peter Lejeune Dirichlet [Dir] and Adrien Marie Legendre [Leg] in 1825. Their elementary arguments are quite involved. (Cf. [Edw], sec. 3.3.) In 1839, Fermat’s equation for exponent 7 also yielded to elementary methods, through the heroic eﬀorts of Gabriel Lam´e. Lam´e’s proof was even more intricate than the proof for exponent 5, and suggested that to go further, new theoretical insights would be needed. The work of Sophie Germain: Around 1820, in a letter to Gauss, Sophie Germain proved that if ` is a prime and q = 2`+1 is also prime, then Fermat’s equation x` + y` = z` with exponent ` has no solutions (x,y,z) with xyz 6= 0 (mod `). Germain’s theorem was the ﬁrst really general proposition on Fermat’s Last Theorem, unlike the previous results which considered the Fermat equation one exponent at a time. The case where the solution (x,y,z) to x` +y` = z` satisﬁes xyz 6= 0 (mod` ) was called the ﬁrst case of Fermat’s Last Theorem, and the case where ` divides xyz, the second case. It was realized at that time that the ﬁrst case was generally easier to handle: Germain’s theorem was extended, using similar ideas, to cases where k`+1 is prime and k is small, and this led to a proof that there were no ﬁrst case solutions to Fermat’s equation with prime exponents ` ≤ 100, which in 1830 represented a signiﬁcant advance. The division between ﬁrst and second case remained fundamental in much of the later work on the subject. In 1977, Terjanian [Te] proved that if the equation x2` +y2` = z2` has a solution (x,y,z), then 2` divides either x or y, i.e., “the ﬁrst case of Fermat’s Last Theorem is true for even exponents”. His simple and elegant proof used only techniques that were available to Germain and her contemporaries. The work of Kummer: The work of Ernst Eduard Kummer marked the beginning of a new era in the study of Fermat’s Last Theorem. For the ﬁrst time, sophisticated concepts of algebraic number theory and the theory of L-functions were brought to bear on a question that had until then been addressed only with elementary methods. While he fell short of providing a complete solution, Kummer made substantial progress. He showed how Fermat’s Last Theorem is intimately tied to deep questions on class numbers of cyclotomic ﬁelds which are still an active subject of research. Kummer’s approach relied on the factorization (x + y)(x + ζ`y)···(x + ζ`−1 ` y) = z` of Fermat’s equation over the ring Z[ζ`] generated by the `th roots of unity. One observes that the greatest common divisor of any two factors in the product on the left divides the element (1−ζ`), which is an element of norm `. 4 Since the product of these numbers is a perfect `-th power, one is tempted to conclude that (x+y),... ,(x+ζ`−1 ` y) are each `-th powers in the ring Z[ζ`] up to units in this ring, and up to powers of (1−ζ`). Such an inference would be valid if one were to replace Z[ζ`] by Z, and is a direct consequence of unique factorization of integers into products of primes. We say that a ring R has property UF if every non-zero element of R is uniquely a product of primes, up to units. Mathematicians such as Lam´e made attempts at proving Fermat’s Last Theorem based on the mistaken assumption that the rings Z[ζ`] had property UF. Legend even has it that Kummer fell into this trap, although this story now has been discredited; see for example [Edw], sec. 4.1. In fact, property UF is far from being satisﬁed in general: one now knows that the rings Z[ζ`] have property UF only for ` < 23 (cf. [Wa], ch. 1). It turns out that the full force of property UF is not really needed in the applications to Fermat’s Last Theorem. Say that a ring R has property UF` if the following inference is valid: ab = z`, and gcd(a,b) = 1 ⇒ a and b are `th powers up to units of R. If a ring R has property UF, then it also has property UF`, but the converse need not be true. Kummer showed that Fermat’s last theorem was true for exponent ` if Z[ζ`] satisﬁed the property UF` (cf. [Wa]). The proof is far from trivial, because of diﬃculties arising from the units in Z[ζ`] as well as from the possible failure of property UF. (A number of Kummer’s contemporaries, such as Cauchy and Lam´e, seem to have overlooked both of these diﬃculties in their attempts to prove Fermat’s Last Theorem.) Kummer then launched a systematic study of the property UF` for the rings Z[ζ`]. He showed that even if Z[ζ`] failed to have unique factorization, it still possessed unique factorization into prime ideals. He deﬁned the ideal class group as the quotient of the group of fractional ideals by its subgroup consisting of principal ideals, and was able to establish the ﬁniteness of this class group. The order of the class group of Z[ζ`], denoted h`, could be taken as a measure of the failure of the ring Z[ζ`] to satisfy UF. It was rather straightforward to show that if ` did not divide h`, then Z[ζ`] satisﬁed the property UF`. In this case, one called ` a regular prime. Kummer thus showed that Fermat’s last theorem is true for exponent ` if ` is a regular prime. He did not stop here. For it remained to give an eﬃcient means of computing h`, or at least an eﬃcient way of checking when ` divides h`. The class number h` can be factorized as a product h` = h+ ` h− ` , 5 where h+ ` is the class number of the real subﬁeld Q(ζ`)+, and h− ` is deﬁned as h`/h+ `. Essentially because of the units in Q(ζ`)+, the factor h+ ` is somewhat diﬃcult to compute, while, because the units in Q(ζ`)+ generate the group of units in Q(ζ`) up to ﬁnite index, the term h− ` can be expressed in a simple closed form. Kummer showed that if ` divides h+ `, then ` divides h− `. Hence, ` divides h` if and only if ` divides h− `. This allowed one to avoid the diﬃculties inherent in the calculation of h+ `. Kummer then gave an elegant formula for h− ` by considering the Bernoulli numbers Bn, which are rational numbers deﬁned by the formula x ex −1 =XBn n! xn. He produced an explicit formula for the class number h− `, and concluded that if ` does not divide the numerator of B2i, for 1 ≤ i ≤ (` − 3)/2, then ` is regular, and conversely. The conceptual explanation for Kummer’s formula for h− ` lies in the work of Dirichlet on the analytic class number formula, where it is shown that h− ` can be expressed as a product of special values of certain (abelian) L-series L(s,χ) = ∞ X n=1 χ(n)n−s associated to odd Dirichlet characters. Such special values in turn can be expressed in terms of certain generalized Bernoulli numbers B1,χ, which are related to the Bernoulli numbers Bi via congruences mod `. (For more details, see [Wa].) These considerations led Kummer to initiate a deep study relating congruence properties of special values of L-functions and of class numbers, which was to emerge as a central concern of modern algebraic number theory, and was to reappear – in a surprisingly diﬀerent guise – at the heart of Wiles’ strategy for proving the Shimura-Taniyama conjecture. Later developments: Kummer’s work had multiple ramiﬁcations, and led to a very active line of enquiry pursued by many people. His formulae relating Bernoulli numbers to class numbers of cyclotomic ﬁelds were reﬁned by Kenneth Ribet [R1], Barry Mazur and Andrew Wiles [MW], using new methods from the theory of modular curves which also play a central role in Wiles’ more recent work. (Later Francisco Thaine [Th] reproved some of the results of Mazur and Wiles using techniques inspired directly from a reading of Kummer.) In a development more directly related to Fermat’s Last Theorem, Wieferich proved that if `2 does not divide 2`−1 −1, then the ﬁrst case of Fermat’s Last Theorem is true for exponent `. (Cf. [Ri], lecture VIII.) 6 There were many other reﬁnements of similar criteria for Fermat’s Last theorem to be true. Computer calculations based on these criteria led to a veriﬁcation that Fermat’s Last theorem is true for all odd prime exponents less than four million [BCEM], and that the ﬁrst case is true for all ` ≤ 8.858·1020 [Su]. The condition that ` is a regular prime seems to hold heuristically for about 61% of the primes. (See the discussion on p. 63, and also p. 108, of [Wa], for example.) In spite of the convincing numerical evidence, it is still not known if there are inﬁnitely many regular primes. Ironically, it is not too diﬃcult to show that there are inﬁnitely many irregular primes. (Cf. [Wa].) Thus the methods introduced by Kummer, after leading to very strong results in the direction of Fermat’s Last theorem, seemed to become mired in diﬃculties, and ultimately fell short of solving Fermat’s conundrum1. Faltings’ proof of the Mordell conjecture: In 1985, Gerd Faltings [Fa] proved the very general statement (which had previously been conjectured by Mordell) that any equation in two variables corresponding to a curve of genus strictly greater than one had (at most) ﬁnitely many rational solutions. In the context of Fermat’s Last Theorem, this led to the proof that for each exponent n ≥ 3, the Fermat equation xn + yn = zn has at most ﬁnitely many integer solutions (up to the obvious rescaling). Andrew Granville [Gra] and Roger Heath-Brown [HB] remarked that Faltings’ result implies Fermat’s Last Theorem for a set of exponents of density one. However, Fermat’s Last Theorem was still not known to be true for an inﬁnite set of prime exponents. In fact, the theorem of Faltings seemed illequipped for dealing with the ﬁner questions raised by Fermat in his margin, namely of ﬁnding a complete list of rational points on all of the Fermat curves xn + yn = 1 simultaneously, and showing that there are no solutions on these curves when n ≥ 3 except the obvious ones. Mazur’s work on Diophantine properties of modular curves: Although it was not realized at the time, the chain of ideas that was to lead to a proof of Fermat’s Last theorem had already been set in motion by Barry Mazur in the mid seventies. The modular curves X0(`) and X1(`) introduced in section 1.2 and 1.5 give rise to another naturally occurring inﬁnite family of Diophantine equations. These equations have certain systematic rational solutions corresponding to the cusps that are deﬁned overQ, and are analogous 1However, W. McCallum has recently introduced a technique, based on the method of Chabauty and Coleman, which suggests new directions for approaching Fermat’s Last Theorem via the cyclotomic theory. An application of McCallum’s method to showing the second case of Fermat’s Last Theorem for regular primes is explained in [Mc]. 7 to the so-called “trivial solutions” of Fermat’s equation. Replacing Fermat curves by modular curves, one could ask for a complete list of all the rational points on the curves X0(`) and X1(`). This problem is perhaps even more compelling than Fermat’s Last Theorem: rational points on modular curves correspond to objects with natural geometric and arithmetic interest, namely, elliptic curves with cyclic subgroups or points of order `. In [Maz1] and [Maz2], B. Mazur gave essentially a complete answer to the analogue of Fermat’s Last Theorem for modular curves. More precisely, he showed that if ` 6= 2,3,5 and 7, (i.e., X1(`) has genus > 0) then the curve X1(`) has no rational points other than the “trivial” ones, namely cusps. He proved analogous results for the curves X0(`) in [Maz2], which implied, in particular, that an elliptic curve over Q with square-free conductor has no rational cyclic subgroup of order ` over Q if ` is a prime which is strictly greater than 7. This result appeared a full ten years before Faltings’ proof of the Mordell conjecture. Frey’s strategy: In 1986, Gerhard Frey had the insight that these constructions might provide a precise link between Fermat’s Last Theorem and deep questions in the theory of elliptic curves, most notably the Shimura Taniyama conjecture. Given a solution a` + b` = c` to the Fermat equation of prime degree `, we may assume without loss of generality that a` ≡−1 (mod 4) and that b` ≡ 0 (mod 32). Frey considered (following Hellegouarch, [He], p. 262; cf. also Kubert-Lang [KL], ch. 8, §2) the elliptic curve E : y2 = x(x−a`)(x + b`). This curve is semistable, i.e., it has square-free conductor. Let E[`] denote the group of points of order ` on E deﬁned over some (ﬁxed) algebraic closure ¯ Q of Q, and let L denote the smallest number ﬁeld over which these points are deﬁned. This extension appears as a natural generalization of the cyclotomic ﬁelds Q(ζ`) studied by Kummer. What singles out the ﬁeld L for special attention is that it has very little ramiﬁcation: using Tate’s analytic description of E at the primes dividing abc, it could be shown that L was ramiﬁed only at 2 and `, and that the ramiﬁcation of L at these two primes was rather restricted. (See theorem 2.15 of section 2.2 for a precise statement.) Moreover, the results of Mazur on the curve X0(`) could be used to show that L is large, in the following precise sense. The space E[`] is a vector space of dimension 2 over the ﬁnite ﬁeld F` with ` elements, and the absolute Galois group GQ = Gal(¯ Q/Q) acts F`-linearly on E[`]. Choosing an F`-basis for E[`], the action is described by a representation ¯ ρE,` : Gal(L/Q) ,→ GL2(F`). 8 Mazur’s results in [Maz1] and [Maz2] imply that ¯ ρE,` is irreducible if ` > 7 (using the fact that E is semi-stable). In fact, combined with earlier results of Serre [Se6], Mazur’s results imply that for ` > 7, the representation ¯ ρE,` is surjective, so that Gal(L/Q) is actually isomorphic to GL2(F`) in this case. Serre’s conjectures: In [Se7], Jean-Pierre Serre made a careful study of mod ` Galois representations ¯ ρ : GQ −→ GL2(F`) (and, more generally, of repre-sentations into GL2(k), where k is any ﬁnite ﬁeld). He was able to make very precise conjectures (see section 3.2) relating these representations to modular forms mod `. In the context of the representations ¯ ρE,` that occur in Frey’s construction, Serre’s conjecture predicted that they arose from modular forms (mod `) of weight two and level two. Such modular forms, which correspond to diﬀerentials on the modular curve X0(2), do not exist because X0(2) has genus 0. Thus Serre’s conjecture implied Fermat’s Last Theorem. The link between ﬁelds with Galois groups contained in GL2(F`) and modular forms mod ` still appears to be very deep, and Serre’s conjecture remains a tantalizing open problem. Ribet’s work: lowering the level: The conjecture of Shimura and Taniyama (cf. section 1.8) provides a direct link between elliptic curves and modular forms. It predicts that the representation ¯ ρE,` obtained from the `-division points of the Frey curve arises from a modular form of weight 2, albeit a form whose level is quite large. (It is the product of all the primes dividing abc, where a` + b` = c` is the putative solution to Fermat’s equation.) Ribet [R5] proved that, if this were the case, then ¯ ρE,` would also be associated with a modular form mod ` of weight 2 and level 2, in the way predicted by Serre’s conjecture. This deep result allowed him to reduce Fermat’s Last Theorem to the Shimura-Taniyama conjecture. Wiles’ work: proof of the Shimura-Taniyama conjecture: In [W3] Wiles proves the Shimura-Taniyama conjecture for semi-stable elliptic curves, providing the ﬁnal missing step and proving Fermat’s Last Theorem. After more than 350 years, the saga of Fermat’s Last theorem has come to a spectacular end. The relation between Wiles’ work and Fermat’s Last Theorem has been very well documented (see, for example, [R8], and the references contained therein). Hence this article will focus primarily on the breakthrough of Wiles [W3] and Taylor-Wiles [TW] which leads to the proof of the Shimura-Taniyama conjecture for semi-stable elliptic curves. From elliptic curves to `-adic representations: Wiles’ opening gambit for proving the Shimura-Taniyama conjecture is to view it as part of the more 9 general problem of relating two-dimensional Galois representations and modular forms. The Shimura-Taniyama conjecture states that if E is an elliptic curve over Q, then E is modular. One of several equivalent deﬁnitions of modularity is that for some integer N there is an eigenform f =Panqn of weight two on Γ0(N) such that #E(Fp) = p + 1−ap for all but ﬁnitely primes p. (By an eigenform, here we mean a cusp form which is a normalized eigenform for the Hecke operators; see section 1 for deﬁnitions.) This conjecture acquires a more Galois theoretic ﬂavour when one considers the two dimensional `-adic representation ρE,` : GQ −→ GL2(Z`) obtained from the action of GQ on the `-adic Tate module of E: T`E = lim ← E[ln](¯ Q). An `-adic representation ρ of GQ is said to arise from an eigenform f =Panqn with integer coeﬃcients an if tr(ρ(Frobp)) = ap, for all but ﬁnitely many primes p at which ρ is unramiﬁed. Here Frob p is a Frobenius element at p (see section 2), and its image under ρ is a well-deﬁned conjugacy class. A direct computation shows that #E(Fp) = p + 1 − tr(ρE,`(Frobp)) for all primes p at which ρE,` is unramiﬁed, so that E is modular (in the sense deﬁned above) if and only if for some `, ρE,` arises from an eigenform. In fact the Shimura-Taniyama conjecture can be generalized to a conjecture that every `-adic representation, satisfying suitable local conditions, arises from a modular form. Such a conjecture was proposed by Fontaine and Mazur