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"page_content='APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31, Number 1, July 1994, Pages 15-38\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n4 9 9 1\\n\\nK. RUBIN AND A. SILVERBERG\\n\\nl u J\\n\\nAbstract. In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence, Fermat’s Last Theorem. This report for nonexperts dis- cusses the mathematics involved in Wiles’ lectures, including the necessary background and the mathematical history.\\n\\n1\\n\\n] T N . h t a m\\n\\nIntroduction\\n\\nOn June 23, 1993, Andrew Wiles wrote on a blackboard, before an audience at the Newton Institute in Cambridge, England, that if p is a prime number, u, v, and w are rational numbers, and up + vp + wp = 0, then uvw = 0. In other words, he announced that he could prove Fermat’s Last Theorem. His announce- ment came at the end of his series of three talks entitled “Modular forms, elliptic curves, and Galois representations” at the week-long workshop on “p-adic Galois representations, Iwasawa theory, and the Tamagawa numbers of motives”.\\n\\n[\\n\\n1 v 0 2 2 7 0 4 9 / h t a m : v i X r a\\n\\nIn the margin of his copy of the works of Diophantus, next to a problem on\\n\\nPythagorean triples, Pierre de Fermat (1601–1665) wrote:\\n\\nCubum autem in duos cubos, aut quadratoquadratum in duos quadrato- quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere : cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.\\n\\n(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)\\n\\nWe restate Fermat’s conjecture as follows.\\n\\nFermat’s Last Theorem. If n > 2, then an +bn = cn has no solutions in nonzero integers a, b, and c.\\n\\nA proof by Fermat has never been found, and the problem has remained open, inspiring many generations of mathematicians. Much of modern number theory has been built on attempts to prove Fermat’s Last Theorem. For details on the\\n\\nReceived by the editors November 29, 1993. 1991 Mathematics Subject Classification. Primary 11G05; Secondary 11D41, 11G18. The authors thank the National Science Foundation for financial support.\\n\\nc(cid:13)1994 American Mathematical Society 0273-0979/94 $1.00 + $.25 per page\\n\\n1\\n\\n2\\n\\nK. RUBIN AND A. SILVERBERG\\n\\nhistory of Fermat’s Last Theorem (last because it is the last of Fermat’s questions to be answered) see [5], [6], and [26].\\n\\nWhat Andrew Wiles announced in Cambridge was that he could prove “many” elliptic curves are modular, sufficiently many to imply Fermat’s Last Theorem. In this paper we will explain Wiles’ work on elliptic curves and its connection with 1 we introduce elliptic curves and modularity, and Fermat’s Last Theorem. give the connection between Fermat’s Last Theorem and the Taniyama-Shimura Conjecture on the modularity of elliptic curves. In 2 we describe how Wiles re- duces the proof of the Taniyama-Shimura Conjecture to what we call the Modular Lifting Conjecture (which can be viewed as a weak form of the Taniyama-Shimura Conjecture), by using a theorem of Langlands and Tunnell. In 4 we show § how the Semistable Modular Lifting Conjecture is related to a conjecture of Mazur on deformations of Galois representations (Conjecture 4.2), and in 5 we describe Wiles’ method of attack on this conjecture. In order to make this survey as acces- sible as possible to nonspecialists, the more technical details are postponed as long as possible, some of them to the appendices.\\n\\nIn\\n\\n§\\n\\n§\\n\\n3 and §\\n\\n§\\n\\nMuch of this report is based on Wiles’ lectures in Cambridge. The authors apol- ogize for any errors we may have introduced. We also apologize to those whose mathematical contributions we, due to our incomplete understanding, do not prop- erly acknowledge.\\n\\nThe ideas Wiles introduced in his Cambridge lectures will have an important influence on research in number theory. Because of the great interest in this subject and the lack of a publicly available manuscript, we hope this report will be useful to the mathematics community. In early December 1993, shortly before this paper went to press, Wiles announced that “the final calculation of a precise upper bound for the Selmer group in the semistable case” (see 5.4 below) “is not yet § complete as it stands,” but that he believes he will be able to finish it in the near future using the ideas explained in his Cambridge lectures. While Wiles’ proof of Theorem 5.3 below and Fermat’s Last Theorem depends on the calculation he referred to in his December announcement, Theorem 5.4 and Corollary 5.5 do not. Wiles’ work provides for the first time infinitely many modular elliptic curves over the rational numbers which are not isomorphic over the complex numbers (see 5.5 for an explicit infinite family).\\n\\n5.3 and\\n\\n§\\n\\n§\\n\\nNotation. The integers, rational numbers, complex numbers, and p-adic integers will be denoted Z, Q, C, and Zp, respectively. If F is a field, then ¯F denotes an algebraic closure of F .\\n\\n1. Connection between Fermat’s Last Theorem and elliptic curves\\n\\n1.1. Fermat’s Last Theorem follows from modularity of elliptic curves. Suppose Fermat’s Last Theorem were false. Then there would exist nonzero integers a, b, c, and n > 2 such that an + bn = cn. It is easy to see that no generality is lost by assuming that n is a prime greater than three (or greater than four million, by [2]; see [14] for n = 3 and 4) and that a and b are relatively prime. Write down the cubic curve:\\n\\ny2 = x(x + an)(x\\n\\nbn).\\n\\n(1)\\n\\n−\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n3\\n\\n1.4 we will explain what it means for an elliptic curve to be modular. Kenneth Ribet [27] proved that if n is a prime greater than three, a, b, and c are nonzero integers, and an + bn = cn, then the elliptic curve (1) is not modular. But the results announced by Wiles imply the following.\\n\\nIn\\n\\n1.3 we will see that such curves are elliptic curves, and in\\n\\n§\\n\\n§\\n\\nTheorem 1.1 (Wiles). If A and B are distinct, nonzero, relatively prime integers, and AB(A\\n\\nB) is divisible by 16, then the elliptic curve\\n\\n−\\n\\ny2 = x(x + A)(x + B)\\n\\nis modular.\\n\\nbn with a, b, c, and n coming from our hypothetical solution to a Fermat equation as above, we see that the conditions of Theorem 1.1 are satisfied since n 5 and one of a, b, and c is even. Thus Theorem 1.1 and Ribet’s result together imply Fermat’s Last Theorem!\\n\\nTaking A = an and B =\\n\\n−\\n\\n≥\\n\\n1.2. History. The story of the connection between Fermat’s Last Theorem and elliptic curves begins in 1955, when Yutaka Taniyama (1927–1958) posed problems which may be viewed as a weaker version of the following conjecture (see [38]).\\n\\nTaniyama-Shimura Conjecture. Every elliptic curve over Q is modular.\\n\\nThe conjecture in the present form was made by Goro Shimura around 1962–64 and has become better understood due to work of Shimura [33–37] and of Andr´e Weil [42] (see also [7]). The Taniyama-Shimura Conjecture is one of the major conjectures in number theory.\\n\\nBeginning in the late 1960s [15–18], Yves Hellegouarch connected Fermat equa- tions an + bn = cn with elliptic curves of the form (1) and used results about Fer- mat’s Last Theorem to prove results about elliptic curves. The landscape changed abruptly in 1985 when Gerhard Frey stated in a lecture at Oberwolfach that elliptic curves arising from counterexamples to Fermat’s Last Theorem could not be mod- ular [11]. Shortly thereafter Ribet [27] proved this, following ideas of Jean-Pierre Serre [32] (see [24] for a survey). In other words, “Taniyama-Shimura Conjecture\\n\\nFermat’s Last Theorem”. Thus, the stage was set. A proof of the Taniyama-Shimura Conjecture (or enough of it to know that elliptic curves coming from Fermat equations are modular) would be a proof of Fermat’s Last Theorem.\\n\\n⇒\\n\\n1.3. Elliptic curves.\\n\\nDefinition. An elliptic curve over Q is a nonsingular curve defined by an equation of the form\\n\\ny2 + a1xy + a3y = x3 + a2x2 + a4x + a6\\n\\n(2)\\n\\nwhere the coefficients ai are integers. The solution ( on the elliptic curve.\\n\\n, ∞\\n\\n) will be viewed as a point\\n\\n∞\\n\\n4\\n\\nK. RUBIN AND A. SILVERBERG\\n\\nRemarks. (i) A singular point on a curve f (x, y) = 0 is a point where both partial derivatives vanish. A curve is nonsingular if it has no singular points.\\n\\n(ii) Two elliptic curves over Q are isomorphic if one can be obtained from the other by changing coordinates x = A2x′ + B, y = A3y′ + Cx′ + D, with A, B, C, D\\n\\nQ and dividing through by A6.\\n\\n∈ (iii) Every elliptic curve over Q is isomorphic to one of the form\\n\\ny2 = x3 + a2x2 + a4x + a6\\n\\nwith integers ai. A curve of this form is nonsingular if and only if the cubic on the right side has no repeated roots.\\n\\nExample. The equation y2 = x(x + 32)(x\\n\\n42) defines an elliptic curve over Q.\\n\\n−\\n\\n1.4. Modularity. Let H denote the complex upper half plane C : Im(z) > 0 } where Im(z) is the imaginary part of z. If N is a positive integer, define a group of matrices\\n\\nz\\n\\n{\\n\\n∈\\n\\na b c d\\n\\nSL2(Z) : c is divisible by N\\n\\n.\\n\\nΓ0(N ) =\\n\\n∈\\n\\n(z) = az+b The group Γ0(N ) acts on H by linear fractional transformations cz+d . (cid:9) (cid:1) The quotient space H/Γ0(N ) is a (noncompact) Riemann surface. It can be com- pleted to a compact Riemann surface, denoted X0(N ), by adjoining a finite set of points called cusps. The cusps are the finitely many equivalence classes of Q ∞} under the action of Γ0(N ) (see Chapter 1 of [35]). The complex points of an elliptic curve can also be viewed as a compact Riemann surface.\\n\\na b c d\\n\\n(cid:8)(cid:0)\\n\\n(cid:1)\\n\\n(cid:0)\\n\\ni\\n\\n∪{\\n\\nDefinition. An elliptic curve E is modular if, for some integer N , there is a holo- morphic map from X0(N ) onto E.\\n\\nExample. It can be shown that there is a (holomorphic) isomorphism from X0(15) onto the elliptic curve y2 = x(x + 32)(x\\n\\n42).\\n\\n−\\n\\nRemark . There are many equivalent definitions of modularity (see II.4.D of [24] and appendix of [22]). In some cases the equivalence is a deep result. For Wiles’ 1.7 proof of Fermat’s Last Theorem it suffices to use only the definition given in below.\\n\\n§\\n\\n§\\n\\n1.5. Semistability.\\n\\nDefinition. An elliptic curve over Q is semistable at the prime q if it is isomorphic to an elliptic curve over Q which modulo q either is nonsingular or has a singu- lar point with two distinct tangent directions. An elliptic curve over Q is called semistable if it is semistable at every prime.\\n\\nExample. The elliptic curve y2 = x(x + 32)(x isomorphic to y2 + xy + y = x3 + x2 x(x + 42)(x\\n\\n42) is semistable because it is − 10, but the elliptic curve y2 =\\n\\n10x\\n\\n−\\n\\n−\\n\\n32) is not semistable (it is not semistable at 2).\\n\\n−\\n\\n2 we explain how Wiles shows that his main result on Galois representations (Theorem 5.3) implies the following part of the Taniyama-Shimura Conjecture.\\n\\nBeginning in\\n\\n§\\n\\nSemistable Taniyama-Shimura Conjecture. Every semistable elliptic curve over Q is modular.\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n5\\n\\nProposition 1.2. The Semistable Taniyama-Shimura Conjecture implies Theorem 1.1.\\n\\nProof. If A and B are distinct, nonzero, relatively prime integers, write EA,B for the elliptic curve defined by y2 = x(x + A)(x + B). Since EA,B and E−A,−B are isomorphic over the complex numbers (i.e., as Riemann surfaces), EA,B is modular if and only if E−A,−B is modular. If further AB(A B) is divisible by 16, then either EA,B or E−A,−B is semistable (this is easy to check directly; see for example I.1 of [24]). The Semistable Taniyama-Shimura Conjecture now implies that both § EA,B and E−A,−B are modular, and thus implies Theorem 1.1.\\n\\n−\\n\\nRemark . In 1.1 we saw that Theorem 1.1 and Ribet’s Theorem together imply Fermat’s Last Theorem. Therefore, the Semistable Taniyama-Shimura Conjecture implies Fermat’s Last Theorem.\\n\\n§\\n\\n1.6. Modular forms. In this paper we will work with a definition of modularity which uses modular forms.\\n\\nDefinition. If N is a positive integer, a modular form f of weight k for Γ0(N ) is C which satisfies a holomorphic function f : H\\n\\n→\\n\\nf (γ(z)) = (cz + d)kf (z)\\n\\na b c d\\n\\nH,\\n\\n(3)\\n\\nΓ0(N ) and z\\n\\nfor every γ =\\n\\n∈\\n\\n∈\\n\\n(cid:1)\\n\\n(cid:0)\\n\\nand is holomorphic at the cusps (see Chapter 2 of [35]).\\n\\n1 1 0 1\\n\\nΓ0(N )), so ∞ n=0 ane2πinz, with complex numbers an and it has a Fourier expansion f (z) = (cid:1) . We say f is a cusp form if it with n vanishes at all the cusps; in particular for a cusp form the coefficient a0 (the value at i\\n\\nA modular form f satisfies f (z) = f (z + 1) (apply (3) to\\n\\n∈\\n\\n(cid:0)\\n\\n0 because f is holomorphic at the cusp i\\n\\n≥\\n\\n∞\\n\\nP\\n\\n) is zero. Call a cusp form normalized if a1 = 1.\\n\\n∞ For fixed N there are commuting linear operators (called Hecke operators) Tm, 1, on the (finite-dimensional) vector space of cusp forms of weight\\n\\nfor integers m two for Γ0(N ) (see Chapter 3 of [35]). If f (z) =\\n\\n≥\\n\\n∞ n=1 ane2πinz, then\\n\\nP danm/d2\\n\\n∞\\n\\ne2πinz\\n\\n(4)\\n\\nTmf (z) =\\n\\nn=1 X\\n\\n(d,N )=1 d|(n,m)\\n\\n(cid:0) X\\n\\n(cid:1)\\n\\nwhere (a, b) denotes the greatest common divisor of a and b and a b means that a divides b. The Hecke algebra T (N ) is the ring generated over Z by these operators.\\n\\n|\\n\\nDefinition. In this paper an eigenform will mean a normalized cusp form of weight two for some Γ0(N ) which is an eigenfunction for all the Hecke operators.\\n\\n∞ n=1 ane2πinz is an eigenform, then Tmf = amf for all m.\\n\\nBy (4), if f (z) =\\n\\nP\\n\\n6\\n\\nK. RUBIN AND A. SILVERBERG\\n\\n1.7. Modularity, revisited. Suppose E is an elliptic curve over Q. If p is a prime, write Fp for the finite field with p elements, and let E(Fp) denote the Fp- solutions of the equation for E (including the point at infinity). We now give a second definition of modularity for an elliptic curve.\\n\\nDefinition. An elliptic curve E over Q is modular if there exists an eigenform\\n\\n∞ n=1 ane2πinz such that for all but finitely many primes q,\\n\\n#(E(Fq)).\\n\\n(5) P\\n\\naq = q + 1\\n\\n− 2. An overview\\n\\nThe flow chart shows how Fermat’s Last Theorem would follow if one knew the Semistable Modular Lifting Conjecture (Conjecture 2.1) for the primes 3 and 5. 1 we discussed the upper arrow, i.e., the implication “Semistable Taniyama- In § Fermat’s Last Theorem”. In this section we will discuss the Shimura Conjecture other implications in the flow chart. The implication given by the lowest arrow is straightforward (Proposition 2.3), while the middle one uses an ingenious idea of Wiles (Proposition 2.4).\\n\\n⇒\\n\\nFermat’s Last Theorem\\n\\n✻\\n\\nSemistable Taniyama-Shimura Conjecture\\n\\n✻\\n\\n(cid:0)\\n\\n❅ ❅\\n\\n(cid:0)\\n\\nSemistable Taniyama-Shimura for ¯ρE,3 irreducible\\n\\nSemistable Modular Lifting for p = 5\\n\\n✻\\n\\n(cid:0) (cid:0)\\n\\n❅\\n\\n❅\\n\\nSemistable Modular Lifting for p = 3\\n\\nLanglands-Tunnell Theorem\\n\\nSemistable Modular Lifting Conjecture\\n\\nFermat’s Last Theorem .\\n\\n⇒\\n\\nRemark . By the Modular Lifting Conjecture we will mean the Semistable Modular Lifting Conjecture with the hypothesis of semistability removed. The arguments of this section can also be used to show that the Modular Lifting Conjecture for p = 3 and 5, together with the Langlands-Tunnell Theorem, imply the full Taniyama- Shimura Conjecture.\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n7\\n\\n2.1. Semistable Modular Lifting. Let ¯Q denote the algebraic closure of Q in C, and let GQ be the Galois group Gal( ¯Q/Q). If p is a prime, write\\n\\nF× p\\n\\n¯εp : GQ\\n\\n→\\n\\nfor the character giving the action of GQ on the p-th roots of unity. For the facts about elliptic curves stated below, see [39]. If E is an elliptic curve over Q and F is a subfield of the complex numbers, there is a natural commutative group law on the set of F -solutions of E, with the point at infinity as the identity element. Denote this group E(F ). If p is a prime, write E[p] for the subgroup of points in E( ¯Q) of order dividing p. Then E[p] ∼= F2 p. The action of GQ on E[p] gives a continuous representation\\n\\nGL2(Fp)\\n\\n¯ρE,p : GQ\\n\\n→\\n\\n(defined up to isomorphism) such that\\n\\n(6)\\n\\ndet(¯ρE,p) = ¯εp\\n\\nand for all but finitely many primes q,\\n\\n#(E(Fq))\\n\\n(7)\\n\\ntrace(¯ρE,p(Frobq))\\n\\nq + 1\\n\\n(mod p).\\n\\n≡ (See Appendix A for the definition of the Frobenius elements Frobq ∈ to each prime number q.)\\n\\n−\\n\\nGQ attached\\n\\n∞ n=1 ane2πinz is an eigenform, let\\n\\nOf denote the ring of integers of the number field Q(a2, a3, . . . ). (Recall that our eigenforms are normalized so that a1 = 1.)\\n\\nIf f (z) =\\n\\nP\\n\\nThe following conjecture is in the spirit of a conjecture of Mazur (see Conjectures\\n\\n3.2 and 4.2).\\n\\nConjecture 2.1 (Semistable Modular Lifting Conjecture). Suppose p is an odd prime and E is a semistable elliptic curve over Q satisfying\\n\\n(a) ¯ρE,p is irreducible, (b) there are an eigenform f (z) =\\n\\n∞ n=1 ane2πinz and a prime ideal λ of\\n\\nOf\\n\\nsuch that p\\n\\nλ and for all but finitely many primes q,\\n\\n∈\\n\\nP\\n\\n#(E(Fq))\\n\\naq ≡\\n\\nq + 1\\n\\n(mod λ).\\n\\n−\\n\\nThen E is modular.\\n\\nThe Semistable Modular Lifting Conjecture is a priori weaker than the Semi- stable Taniyama-Shimura Conjecture because of the extra hypotheses (a) and (b). The more serious condition is (b); there is no known way to produce such a form in general. But when p = 3, the existence of such a form follows from the theorem below of Tunnell [41] and Langlands [20]. Wiles then gets around condition (a) by a clever argument (described below) which, when ¯ρE,3 is not irreducible, allows him to use p = 5 instead.\\n\\n8\\n\\nK. RUBIN AND A. SILVERBERG\\n\\n2.2. Langlands-Tunnell Theorem. In order to state the Langlands-Tunnell Theorem, we need weight-one modular forms for a subgroup of Γ0(N ). Let\\n\\na b c d\\n\\nSL2(Z) : c\\n\\n0 (mod N ), a\\n\\nd\\n\\n1 (mod N )\\n\\n.\\n\\nΓ1(N ) =\\n\\n∈\\n\\n≡\\n\\n≡\\n\\n≡\\n\\n(cid:1)\\n\\n(cid:9)\\n\\n(cid:8)(cid:0)\\n\\nReplacing Γ0(N ) by Γ1(N ) in 1.6, one can define the notion of cusp forms on § Γ1(N ). See Chapter 3 of [35] for the definitions of the Hecke operators on the space of weight-one cusp forms for Γ1(N ).\\n\\nTheorem 2.2 (Langlands-Tunnell). Suppose ρ : GQ GL2(C) is a continuous irreducible representation whose image in PGL2(C) is a subgroup of S4 (the sym- metric group on four elements ), τ is complex conjugation, and det(ρ(τ )) = 1. ∞ n=1 bne2πinz for some Γ1(N ), which is an Then there is a weight-one cusp form eigenfunction for all the corresponding Hecke operators, such that for all but finitely many primes q,\\n\\n→\\n\\n−\\n\\nP\\n\\n(8)\\n\\nbq = trace(ρ(Frobq)).\\n\\nThe theorem as stated by Langlands [20] and by Tunnell [41] produces an auto- morphic representation rather than a cusp form. Using the fact that det(ρ(τ )) = 1, standard techniques (see for example [12]) show that this automorphic repre-\\n\\n− sentation corresponds to a weight-one cusp form as in Theorem 2.2.\\n\\n2.3. Semistable Modular Lifting\\n\\nSemistable Taniyama-Shimura.\\n\\n⇒\\n\\nProposition 2.3. Suppose the Semistable Modular Lifting Conjecture is true for p = 3, E is a semistable elliptic curve, and ¯ρE,3 is irreducible. Then E is modular.\\n\\nProof. It suffices to show that hypothesis (b) of the Semistable Modular Lifting Conjecture is satisfied with the given curve E, for p = 3. There is a faithful representation\\n\\nGL2(Z[√\\n\\nGL2(C)\\n\\nψ : GL2(F3) ֒\\n\\n2])\\n\\n−\\n\\n⊂\\n\\n→\\n\\nGL2(F3),\\n\\nsuch that for every g\\n\\n∈ trace(ψ(g))\\n\\n(mod(1 + √\\n\\n(9)\\n\\ntrace(g)\\n\\n2))\\n\\n≡\\n\\n−\\n\\nand\\n\\n(10)\\n\\ndet(ψ(g))\\n\\ndet(g)\\n\\n(mod 3).\\n\\n≡\\n\\nExplicitly, ψ can be defined on generators of GL2(F3) by\\n\\n√\\n\\n1 1 1 0\\n\\n1 1 1 0\\n\\n1 1\\n\\n1 1\\n\\n2 1 1 0\\n\\n.\\n\\nψ\\n\\n=\\n\\nand ψ\\n\\n=\\n\\n− −\\n\\n− −\\n\\n−\\n\\n−\\n\\n(cid:19)\\n\\n(cid:18)(cid:18)\\n\\n(cid:19)(cid:19)\\n\\n(cid:18)\\n\\n(cid:18)(cid:18) ¯ρE,3. If τ is complex conjugation, then it follows from (6) and (10) that 1. The image of ψ in PGL2(C) is a subgroup of PGL2(F3) ∼= S4.\\n\\n(cid:19)\\n\\n(cid:19)(cid:19)\\n\\n(cid:18)\\n\\nLet ρ = ψ ◦ det(ρ(τ )) = Using that ¯ρE,3 is irreducible, one can show that ρ is irreducible.\\n\\n−\\n\\n∞ n=1 bne2πinz be a weight-one cusp form for some Γ1(N ) obtained by applying the Langlands-Tunnell\\n\\nLet p be a prime of ¯Q containing 1 + √\\n\\n2. Let g(z) =\\n\\n−\\n\\nP\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n9\\n\\nTheorem (Theorem 2.2) to ρ. It follows from (6) and (10) that N is divisible by 3. The function\\n\\n0 if d 1 if d 1 if d\\n\\n0 (mod 3), 1 (mod 3), 2 (mod 3)\\n\\n∞\\n\\n≡ ≡ ≡\\n\\nχ(d)e2πinz where χ(d) =\\n\\nE(z) = 1 + 6\\n\\n\\uf8f1 \\uf8f2\\n\\nn=1 X\\n\\nXd|n\\n\\n−\\n\\n∞ n=1 cne2πinz is a weight-one modular form for Γ1(3). The product g(z)E(z) = It is now is a weight-two cusp form for Γ0(N ) with cn ≡ bn possible to find an eigenform f (z) = (mod p) for every n (see 6.10 and 6.11 of [4]). By (7), (8), and (9), f satisfies (b) of the Semistable Modular Lifting Conjecture with p = 3 and with λ = p\\n\\n\\uf8f3\\n\\nbn (mod p) for all n. P n=1 ane2πinz on Γ0(N ) such that an ≡ ∩ Of .\\n\\n∞\\n\\nP\\n\\nProposition 2.4 (Wiles). Suppose the Semistable Modular Lifting Conjecture is true for p = 3 and 5, E is a semistable elliptic curve over Q, and ¯ρE,3 is reducible. Then E is modular.\\n\\nProof. The elliptic curves over Q for which both ¯ρE,3 and ¯ρE,5 are reducible are all known to be modular (see Appendix B.1). Thus we can suppose ¯ρE,5 is irreducible. It suffices to produce an eigenform as in (b) of the Semistable Modular Lifting Conjecture, but this time there is no analogue of the Langlands-Tunnell Theorem to help. Wiles uses the Hilbert Irreducibility Theorem, applied to a parameter space of elliptic curves, to produce another semistable elliptic curve E′ over Q satisfying\\n\\n(i) ¯ρE′,5 is isomorphic to ¯ρE,5, and (ii) ¯ρE′,3 is irreducible.\\n\\n(In fact there will be infinitely many such E′; see Appendix B.2.) Now by Proposi- ∞ n=1 ane2πinz be a corresponding eigenform. tion 2.3, E′ is modular. Let f (z) = Then for all but finitely many primes q, P\\n\\n#(E′(Fq)) trace(¯ρE,5(Frobq))\\n\\naq = q + 1\\n\\ntrace(¯ρE′,5(Frobq)) #(E(Fq)) q + 1\\n\\n−\\n\\n≡ ≡\\n\\n(mod 5)\\n\\n≡\\n\\n−\\n\\nby (7). Thus the form f satisfies hypothesis (b) of the Semistable Modular Lifting Conjecture, and we conclude that E is modular.\\n\\nTaken together, Propositions 2.3 and 2.4 show that the Semistable Modular Lifting Conjecture for p = 3 and 5 implies the Semistable Taniyama-Shimura Con- jecture.\\n\\n3. Galois representations\\n\\nThe next step is to translate the Semistable Modular Lifting Conjecture into a conjecture (Conjecture 3.2) about the modularity of liftings of Galois repre- sentations. Throughout this paper, if A is a topological ring, a representation GL2(A) will mean a continuous homomorphism and [ρ] will denote the ρ : GQ isomorphism class of ρ. If p is a prime, let\\n\\n→\\n\\nZ× p\\n\\nεp : GQ\\n\\n→\\n\\nbe the character giving the action of GQ on p-power roots of unity.\\n\\n10\\n\\nK. RUBIN AND A. SILVERBERG\\n\\n3.1. The p-adic representation attached to an elliptic curve. Suppose E is an elliptic curve over Q and p is a prime number. For every positive integer n, write E[pn] for the subgroup in E( ¯Q) of points of order dividing pn and Tp(E) for the inverse limit of the E[pn] with respect to multiplication by p. For every n, E[pn] ∼= (Z/pnZ)2, and so Tp(E) ∼= Z2 p. The action of GQ induces a representation\\n\\nGL2(Zp)\\n\\nρE,p : GQ\\n\\n→\\n\\nsuch that det(ρE,p) = εp and for all but finitely many primes q,\\n\\n#(E(Fq)).\\n\\n(11)\\n\\ntrace(ρE,p(Frobq)) = q + 1\\n\\n−\\n\\nComposing ρE,p with the reduction map from Zp to Fp gives ¯ρE,p of\\n\\n2.1. §\\n\\n3.2. Modular representations. If f is an eigenform and λ is a prime ideal of Of at λ. Of , let\\n\\nOf,λ denote the completion of\\n\\nDefinition. If A is a ring, a representation ρ : GQ if there are an eigenform f (z) = homomorphism ι :\\n\\nGL2(A) is called modular ∞ n=1 ane2πinz, a ring A′ containing A, and a\\n\\n→\\n\\nA′ such that for all but finitely many primes q,\\n\\nOf →\\n\\nP\\n\\ntrace(ρ(Frobq)) = ι(aq).\\n\\n∞ n=1 ane2πinz and a prime ideal λ of\\n\\nExamples. (i) Given an eigenform f (z) = Of , Eichler and Shimura (see\\n\\n7.6 of [35]) constructed a representation\\n\\n§\\n\\nP\\n\\nρf,λ : GQ\\n\\nGL2(\\n\\nOf,λ)\\n\\n→\\n\\nZ = pZ) and for all but finitely many primes q,\\n\\nsuch that det(ρf,λ) = εp (where λ\\n\\n∩\\n\\n(12)\\n\\ntrace(ρf,λ(Frobq)) = aq.\\n\\nThus ρf,λ is modular with ι taken to be the inclusion of\\n\\nOf in\\n\\nOf,λ.\\n\\n(ii) Suppose p is a prime and E is an elliptic curve over Q. If E is modular, then ρE,p and ¯ρE,p are modular by (11), (7), and (5). Conversely, if ρE,p is modular, then it follows from (11) that E is modular. This proves the following.\\n\\nTheorem 3.1. Suppose E is an elliptic curve over Q. Then\\n\\nE is modular\\n\\nρE,p is modular for every p\\n\\nρE,p is modular for one p.\\n\\n⇔\\n\\n⇔\\n\\nRemark . In this language, the Semistable Modular Lifting Conjecture says that if p is an odd prime, E is a semistable elliptic curve over Q, and ¯ρE,p is modular and irreducible, then ρE,p is modular.\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n11\\n\\n3.3. Liftings of Galois representations. Fix a prime p and a finite field k of characteristic p. Recall that ¯k denotes an algebraic closure of k.\\n\\nGiven a map φ : A\\n\\nB, the induced map from GL2(A) to GL2(B) will also be\\n\\n→\\n\\ndenoted φ. If ρ : GQ A′ for the composition of ρ with the inclusion of GL2(A) in GL2(A′).\\n\\nGL2(A) is a representation and A′ is a ring containing A, we write\\n\\n→\\n\\nρ\\n\\n⊗\\n\\nDefinition. If ¯ρ : GQ ρ : GQ Zp-algebra and there exists a homomorphism ι : A\\n\\nGL2(k) is a representation, we say that a representation GL2(A) is a lifting of ¯ρ (to A) if A is a complete noetherian local\\n\\n→\\n\\n→\\n\\n¯k such that the diagram\\n\\n→ GL2(A)\\n\\n✟✟✯\\n\\n[ρ]\\n\\n✟✟\\n\\nι ❄ GL2(¯k)\\n\\n✲\\n\\nGQ\\n\\n[ ¯ρ ⊗ ¯k]\\n\\n¯k].\\n\\ncommutes, in the sense that [ι\\n\\nρ] = [¯ρ\\n\\n\\n\\n⊗\\n\\nExamples. (i) If E is an elliptic curve then ρE,p is a lifting of ¯ρE,p.\\n\\n(ii) If E is an elliptic curve, p is a prime, and hypotheses (a) and (b) of Conjecture\\n\\n2.1 hold with an eigenform f and prime ideal λ, then ρf,λ is a lifting of ¯ρE,p.\\n\\n3.4. Deformation data. We will be interested not in all liftings of a given ¯ρ, but rather in those satisfying various restrictions. See Appendix A for the definition of GQ associated to primes q. We say that a representation ρ the inertia groups Iq ⊂ of GQ is unramified at a prime q if ρ(Iq) = 1. If Σ is a set of primes, we say ρ is unramified outside of Σ if ρ is unramified at every q / ∈\\n\\nΣ.\\n\\nDefinition. By deformation data we mean a pair\\n\\n= (Σ, t)\\n\\nD where Σ is a finite set of primes and t is one of the words ordinary or flat.\\n\\nZ×\\n\\nA× be the composition of the\\n\\nIf A is a Zp-algebra, let εA : GQ\\n\\np →\\n\\n→\\n\\ncyclotomic character εp with the structure map.\\n\\nDefinition. Given deformation data type- outside of Σ, and ρ is t at p (where t\\n\\nGL2(A) is if A is a complete noetherian local Zp-algebra, det(ρ) = εA, ρ is unramified\\n\\n, a representation ρ : GQ\\n\\nD\\n\\n→\\n\\nD\\n\\nordinary, flat }\\n\\n; see Appendix C).\\n\\n∈ {\\n\\nDefinition. A representation ¯ρ : GQ eigenform f and a prime ideal λ of\\n\\nmodular if there are an\\n\\nGL2(k) is Of such that ρf,λ is a type-\\n\\n→\\n\\nD\\n\\nlifting of ¯ρ.\\n\\nD\\n\\nRemarks. (i) A representation with a type- fore if a representation is\\n\\nlifting must itself be type-\\n\\n. There-\\n\\nD\\n\\nD and modular.\\n\\nmodular, then it is both type-\\n\\nD\\n\\nD\\n\\n(ii) Conversely, if ¯ρ is type-\\n\\n, modular, and satisfies (ii) of Theorem 5.3 below, -modular, by work of Ribet and others (see [28]). This plays an important\\n\\nD\\n\\nthen ¯ρ is D role in Wiles’ work.\\n\\n12\\n\\nK. RUBIN AND A. SILVERBERG\\n\\n3.5. Mazur Conjecture.\\n\\nDefinition. A representation ¯ρ : GQ ¯ρ\\n\\nGL2(k) is called absolutely irreducible if\\n\\n→\\n\\n¯k is irreducible.\\n\\n⊗\\n\\nThe following variant of a conjecture of Mazur (see Conjecture 18 of [23]; see\\n\\nalso Conjecture 4.2 below) implies the Semistable Modular Lifting Conjecture.\\n\\nConjecture 3.2 (Mazur). Suppose p is an odd prime, k is a finite field of charac- GL2(k) is an absolutely irreducible teristic p, lifting of ¯ρ to the ring of integers of\\n\\nis deformation data, and ¯ρ : GQ -modular representation. Then every type-\\n\\nD\\n\\n→ D\\n\\nD a finite extension of Qp is modular.\\n\\nRemark . Loosely speaking, Conjecture 3.2 says that if ¯ρ is modular, then every lifting which “looks modular” is modular.\\n\\nDefinition. An elliptic curve E over Q has good (respectively, bad ) reduction at a prime q if E is nonsingular (respectively, singular) modulo q. An elliptic curve E over Q has ordinary (respectively, supersingular) reduction at q if E has good reduction at q and E[q] has (respectively, does not have) a subgroup of order q stable under the inertia group Iq.\\n\\nProposition 3.3. Conjecture 3.2 implies Conjecture 2.1.\\n\\nProof. Suppose p is an odd prime and E is a semistable elliptic curve over Q which satisfies (a) and (b) of Conjecture 2.1. We will apply Conjecture 3.2 with ¯ρ = ¯ρE,p. Write τ for complex conjugation. Then τ 2 = 1, and by (6), det(¯ρE,p(τ )) = 1. Since ¯ρE,p is irreducible and p is odd, a simple linear algebra argument now shows that ¯ρE,p is absolutely irreducible.\\n\\n−\\n\\nSince E satisfies (b) of Conjecture 2.1, ¯ρE,p is modular. Let\\n\\nΣ = t = ordinary if E has ordinary or bad reduction at p, t = flat if E has supersingular reduction at p,\\n\\np\\n\\nprimes q : E has bad reduction at q\\n\\n,\\n\\n•\\n\\n{\\n\\n} ∪ {\\n\\n}\\n\\n= (Σ, t).\\n\\nD\\n\\nUsing the semistability of E, one can show that ρE,p is a type- (by combining results of several people; see [28]) that ¯ρE,p is 3.2 then says ρE,p is modular. By Theorem 3.1, E is modular.\\n\\nlifting of ¯ρE,p and -modular. Conjecture\\n\\nD\\n\\nD\\n\\n4. Mazur’s deformation theory\\n\\nNext we reformulate Conjecture 3.2 as a conjecture (Conjecture 4.2) that the algebras which parametrize liftings and modular liftings of a given representation are isomorphic. It is this form of Mazur’s conjecture that Wiles attacks directly.\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n13\\n\\n4.1. The universal deformation algebra R. Fix an odd prime p, a finite field k of characteristic p, deformation data representation ¯ρ : GQ extension of Qp with residue field k.\\n\\n, and an absolutely irreducible type-\\n\\nD\\n\\nD is the ring of integers of a finite\\n\\nGL2(k). Suppose\\n\\n→\\n\\nO\\n\\nDefinition. We say ρ : GQ complete noetherian local commutes\\n\\n)-lifting of ¯ρ if ρ is type-\\n\\n, A is a → -algebra with residue field k, and the following diagram\\n\\nGL2(A) is a (\\n\\n,\\n\\nD\\n\\nO\\n\\nD\\n\\nO\\n\\nGL2(A)\\n\\n✟✟✯\\n\\n[ρ]\\n\\n✟✟\\n\\n❄ GL2(k)\\n\\n✲\\n\\nGQ\\n\\n[ ¯ρ]\\n\\nwhere the vertical map is reduction modulo the maximal ideal of A.\\n\\nTheorem 4.1 (Mazur-Ramakrishna). With p, k, an that for every ( φρ : R\\n\\nas above, there are D GL2(R) of ¯ρ, with the property -algebra homomorphism\\n\\n, ¯ρ, and\\n\\nO\\n\\nalgebra R and a (\\n\\n)-lifting ρR : GQ )-lifting ρ of ¯ρ to A there is a unique\\n\\n,\\n\\nO\\n\\nD\\n\\nO\\n\\n→\\n\\n,\\n\\nD\\n\\nO\\n\\nO\\n\\nA such that the diagram\\n\\n→\\n\\n[ρR]\\n\\n✲\\n\\nGQ\\n\\nGL2(R)\\n\\n❍\\n\\n❍❍\\n\\nφρ ❄ GL2(A)\\n\\n[ρ]\\n\\n❍❍❥\\n\\ncommutes.\\n\\nThis theorem was proved by Mazur [21] in the case when\\n\\nis ordinary and is flat. Theorem 4.1 determines R and ρR up to\\n\\nD\\n\\nby Ramakrishna [25] when isomorphism.\\n\\nD\\n\\n4.2. The universal modular deformation algebra T. Fix an odd prime p, a , and an absolutely irreducible finite field k of characteristic p, deformation data -modular, and fix an type- representation ¯ρ : GQ eigenform f and a prime ideal λ of lifting of ¯ρ. is the ring of integers of a finite extension of Qp with Suppose in addition that residue field k, Of,λ ⊆ O\\n\\nD\\n\\nGL2(k). Assume ¯ρ is\\n\\nD\\n\\n→\\n\\nD\\n\\nOf such that ρf,λ is a type-\\n\\nD\\n\\nO , and the diagram\\n\\nGL2(\\n\\nOf,λ) ❄ GL2(k)\\n\\n✟✟✟✯ ✲\\n\\n[ρf,λ] ✟\\n\\nGQ\\n\\n[ ¯ρ]\\n\\ncommutes, where the vertical map is the reduction map.\\n\\n)-lifting of ¯ρ, and Wiles constructs a generalized Hecke algebra T which has the following properties (recall that Hecke algebras T (N ) were defined in\\n\\nUnder these assumptions ρf,λ ⊗ O 1.6).\\n\\nis a (\\n\\n,\\n\\nD\\n\\nO\\n\\n§\\n\\n(T1) T is a complete noetherian local\\n\\nalgebra with residue field k.\\n\\nO\\n\\n14\\n\\nK. RUBIN AND A. SILVERBERG\\n\\n(T2) There are an integer N divisible only by primes in Σ and a homomorphism by the Σ. By abuse of notation\\n\\nfrom the Hecke algebra T (N ) to T such that T is generated over images of the Hecke operators Tq for primes q / ∈ we write Tq also for its image in T.\\n\\nO\\n\\n(T3) There is a (\\n\\n,\\n\\n)-lifting\\n\\nD\\n\\nO\\n\\nGL2(T)\\n\\nρT : GQ\\n\\n→\\n\\nof ¯ρ with the property that trace(ρT(Frobq)) = Tq for every prime q / ∈\\n\\nΣ. )-lifting of ¯ρ to A, then there is a unique\\n\\n(T4) If ρ is modular and is a (\\n\\n,\\n\\nD\\n\\nO\\n\\nalgebra homomorphism ψρ : T\\n\\nA such that the diagram\\n\\nO\\n\\n→ [ρ T]\\n\\n✲\\n\\nGL2(T)\\n\\nGQ\\n\\n❍\\n\\n❍❍\\n\\nψρ ❄ GL2(A)\\n\\n[ρ]\\n\\n❍❍❥\\n\\ncommutes.\\n\\nSince ρT is a (\\n\\n,\\n\\n)-lifting of ¯ρ, by Theorem 4.1 there is a homomorphism\\n\\nD\\n\\nO\\n\\nT\\n\\nϕ : R\\n\\n→\\n\\nρR. By (T3), ϕ(trace(ρR(Frobq))) = Tq for every\\n\\nsuch that ρT is isomorphic to ϕ prime q / ∈\\n\\nΣ, so it follows from (T2) that ϕ is surjective.\\n\\n4.3. Mazur Conjecture, revisited. Conjecture 3.2 can be reformulated in the following way.\\n\\nConjecture 4.2 (Mazur). Suppose p, k, T is an isomorphism. above map ϕ : R\\n\\n, ¯ρ, and\\n\\nare as in\\n\\n4.2. Then the\\n\\nD\\n\\nO\\n\\n§\\n\\n→\\n\\nConjecture 4.2 was stated in [23] (Conjecture 18) for\\n\\nordinary, and Wiles\\n\\nD\\n\\nmodified the conjecture to include the flat case.\\n\\nProposition 4.3. Conjecture 4.2 implies Conjecture 3.2.\\n\\nProof. Suppose ¯ρ : GQ -modular, A is D the ring of integers of a finite extension of Qp, and ρ is a type- lifting of ¯ρ to A. to be the ring of integers of a sufficiently large finite extension of Qp, and Taking and its residue field, respectively, we may assume that ρ is extending ρ and ¯ρ to A, with φρ a ( as in Theorem 4.1. By (T3) and Theorem 4.1, ψ(Tq) = trace(ρ(Frobq)) for all but 3.5 of [35], given such a homomorphism ψ (and viewing A as finitely many q. By ∞ n=1 ane2πinz where aq = ψ(Tq) for all but a subring of C), there is an eigenform finitely many primes q. Thus ρ is modular.\\n\\nGL2(k) is absolutely irreducible and\\n\\n→\\n\\nD\\n\\nO )-lifting of ¯ρ. Assuming Conjecture 4.2, let ψ = φρ ◦\\n\\nO\\n\\nϕ−1 : T\\n\\n,\\n\\nD\\n\\nO\\n\\n→\\n\\n§\\n\\nP\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n15\\n\\n5. Wiles’ approach to the Mazur Conjecture\\n\\nIn this section we sketch the major ideas of Wiles’ attack on Conjecture 4.2. The first step (Theorem 5.2), and the key to Wiles’ proof, is to reduce Conjecture 4.2 to a bound on the order of the cotangent space at a prime of R. In 5.2 we § see that the corresponding tangent space is a Selmer group, and in 5.3 we outline a general procedure due to Kolyvagin for bounding sizes of Selmer groups. The input for Kolyvagin’s method is known as an Euler system. The most difficult 5.4), and the part described as “not yet complete” in his part of Wiles’ work ( § December announcement, is his construction of a suitable Euler system. In 5.5 we state the results announced by Wiles (Theorems 5.3 and 5.4 and Corollary 5.5) and explain why Theorem 5.3 suffices for proving the Semistable Taniyama-Shimura Conjecture. As an application of Corollary 5.5 we write down an infinite family of modular elliptic curves. , ¯ρ, 5 fix p, k,\\n\\n§\\n\\n§\\n\\n∞ n=1 ane2πinz, and λ as in\\n\\n4.2.\\n\\nFor O By property (T4) there is a homomorphism\\n\\n, f (z) =\\n\\n§\\n\\n§\\n\\nD\\n\\nP\\n\\nπ : T\\n\\n→ O . By property (T2) and (12), π satisfies\\n\\nsuch that π π(Tq) = aq for all but finitely many q.\\n\\nρT is isomorphic to ρf,λ ⊗ O\\n\\n\\n\\n5.1. Key reduction. Wiles uses the following generalization of a theorem of Mazur, which says that T is Gorenstein.\\n\\nTheorem 5.1. There is a (noncanonical ) T-module isomorphism\\n\\n) ∼ →\\n\\nHomO(T,\\n\\nT.\\n\\nO\\n\\nLet η denote the ideal of\\n\\ngenerated by the image under the composition\\n\\nO HomO(T,\\n\\n) ∼ →\\n\\nT π\\n\\nO\\n\\n→ O\\n\\nHomO(T,\\n\\nof the element π ∈ choice of isomorphism in Theorem 5.1.\\n\\n). The ideal η is well defined independent of the\\n\\nO\\n\\nThe map π determines distinguished prime ideals of T and R,\\n\\nϕ) = ϕ−1(pT).\\n\\npT = ker(π),\\n\\npR = ker(π\\n\\n\\n\\nTheorem 5.2 (Wiles). If\\n\\n#(pR/p2\\n\\nR)\\n\\n#(\\n\\n/η) <\\n\\n, ∞\\n\\n≤\\n\\nO\\n\\nT is an isomorphism.\\n\\nthen ϕ : R\\n\\n→\\n\\nThe proof is entirely commutative algebra. The surjectivity of ϕ shows that /η). Thus if\\n\\n#(pR/p2 #(pR/p2\\n\\n#(pT/p2 #(\\n\\nT), and Wiles proves that #(pT/p2\\n\\nR) R)\\n\\nT)\\n\\n#(\\n\\n≥ ≤\\n\\n≥\\n\\nO\\n\\n/η), then\\n\\nO\\n\\n#(pR/p2\\n\\nR) = #(pT/p2\\n\\n(13)\\n\\nT) = #(\\n\\n/η).\\n\\nO\\n\\nThe first equality in (13) shows that ϕ induces an isomorphism of tangent spaces. Wiles uses the second equality in (13) and Theorem 5.1 to deduce that T is a local\\n\\n16\\n\\nK. RUBIN AND A. SILVERBERG\\n\\ncomplete intersection over that\\n\\n(that is, there are f1, . . . , fr ∈ O\\n\\n[[x1, . . . , xr]] such\\n\\nO\\n\\nT ∼=\\n\\n[[x1, . . . , xr]]/(f1, . . . , fr)\\n\\nO\\n\\nas morphism.\\n\\nalgebras). Wiles then combines these two results to prove that ϕ is an iso-\\n\\nO\\n\\n5.2. Selmer groups. In general, if M is a torsion GQ-module, a Selmer group attached to M is a subgroup of the Galois cohomology group H 1(GQ, M ) deter- mined by certain “local conditions” in the following way. If q is a prime with decomposition group Dq ⊂\\n\\nGQ, then there is a restriction map\\n\\nresq : H 1(GQ, M )\\n\\nH 1(Dq, M ).\\n\\n→ Jq ⊆\\n\\nH 1(Dq, M ) : q prime\\n\\n= For a fixed collection of subgroups { the particular problem under consideration, the corresponding Selmer group is\\n\\ndepending on\\n\\nJ\\n\\n}\\n\\nres−1\\n\\nH 1(GQ, M ).\\n\\nS(M ) =\\n\\nq (Jq)\\n\\n⊆\\n\\nq \\\\ Write H i(Q, M ) for H i(GQ, M ), and H i(Qq, M ) for H i(Dq, M ).\\n\\nExample. The original examples of Selmer groups come from elliptic curves. Fix an elliptic curve E and a positive integer m, and take M = E[m], the subgroup of points in E( ¯Q) of order dividing m. There is a natural inclusion\\n\\nH 1(Q, E[m])\\n\\nE(Q)/mE(Q) ֒\\n\\n(14)\\n\\n→\\n\\nE( ¯Q) is any\\n\\nE(Q) to the cocycle σ\\n\\nobtained by sending x point satisfying my = x. Similarly, for every prime q there is a natural inclusion\\n\\nσ(y)\\n\\ny, where y\\n\\n∈\\n\\n7→\\n\\n−\\n\\n∈\\n\\nH 1(Qq, E[m]).\\n\\nE(Qq)/mE(Qq) ֒\\n\\n→ Define the Selmer group S(E[m]) in this case by taking the group Jq to be the image of E(Qq)/mE(Qq) in H 1(Qq, E[m]), for every q. This Selmer group is an important tool in studying the arithmetic of E because it contains (via (14)) E(Q)/mE(Q).\\n\\n5, let m denote the maximal ideal /mn) can be\\n\\nRetaining the notation from the beginning of\\n\\n§\\n\\nand fix a positive integer n. The tangent space HomO(pR/p2 R,\\n\\nof identified with a Selmer group as follows. Let Vn be the matrix algebra M2(\\n\\nO\\n\\nO\\n\\n/mn), with GQ acting via the adjoint repre-\\n\\nO\\n\\nsentation σ(B) = ρf,λ(σ)Bρf,λ(σ)−1. There is a natural injection\\n\\ns : HomO(pR/p2 R,\\n\\n/mn) ֒\\n\\nH 1(Q, Vn)\\n\\nO\\n\\n→\\n\\nwhich is described in Appendix D (see also\\n\\n1.6 of [21]). Wiles defines a collection . Let SD(Vn) denote the associated Selmer\\n\\n§\\n\\nH 1(Qq, Vn) }\\n\\n=\\n\\nJq ⊆\\n\\ndepending on\\n\\nJ group. Wiles proves that s induces an isomorphism\\n\\n{\\n\\nD\\n\\n/mn) ∼ →\\n\\nHomO(pR/p2 R,\\n\\nSD(Vn).\\n\\nO\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n17\\n\\n5.3. Euler systems. We have now reduced the proof of Mazur’s conjecture to bounding the size of the Selmer groups SD(Vn). About five years ago Kolyvagin [19], building on ideas of his own and of Thaine [40], introduced a revolutionary new method for bounding the size of a Selmer group. This new machinery, which is crucial for Wiles’ proof, is what we now describe.\\n\\nH 1(Qq,M ) is } 5.2. Let ˆM = a system of subgroups with associated Selmer group S(M ) as in Hom(M, µm), where µm is the group of m-th roots of unity. For every prime q, the cup product gives a nondegenerate Tate pairing\\n\\nSuppose M is a GQ-module of odd exponent m and\\n\\n=\\n\\nJq ⊆ §\\n\\nJ\\n\\n{\\n\\nH 2(Qq, µm) ∼ → H 1(Q, ˆM ), then\\n\\nH 1(Qq, ˆM )\\n\\niq : H 1(Qq, M )\\n\\nZ/mZ\\n\\n,\\n\\nh\\n\\n×\\n\\n→\\n\\nH 1(Q, M ) and d\\n\\n(see Chapters VI and VII of [3]). If c\\n\\n∈\\n\\n∈\\n\\n(15)\\n\\nresq(c), resq(d) h\\n\\niq = 0.\\n\\nq X\\n\\nH 1(Q, ˆM ) be the Selmer\\n\\nis a finite set of primes. Let S∗\\n\\nSuppose that\\n\\nL ⊆ H 1(Qq, ˆM ) }\\n\\nL group given by the local conditions\\n\\n∗ =\\n\\nJ ∗ q ⊆\\n\\n, where\\n\\nJ\\n\\n{\\n\\nthe orthogonal complement of Jq under H 1(Qq, ˆM )\\n\\n,\\n\\nif q / if q\\n\\n, ∈ L . ∈ L\\n\\niq\\n\\nJ ∗ q =\\n\\nh\\n\\n(\\n\\nH 1(Q, ˆM ), define\\n\\nIf d\\n\\n∈\\n\\nZ/mZ\\n\\nθd :\\n\\nJq →\\n\\nYq∈L\\n\\nby\\n\\nθd((cq)) =\\n\\ncq, resq(d) h\\n\\niq.\\n\\nXq∈L\\n\\nWrite resL : H 1(Q, M ) maps. By (15) and the definition of J ∗ in addition resL is injective on S(M ), then\\n\\nq∈L H 1(Qq, M ) for the product of the restriction ker(θd). If\\n\\n→\\n\\nS∗\\n\\nq , if d\\n\\nL, then resL(S(M ))\\n\\n∈\\n\\n⊆\\n\\nQ\\n\\n#(S(M ))\\n\\n#\\n\\nker(θd)\\n\\n.\\n\\n≤\\n\\n(cid:0) \\\\d∈S∗\\n\\nL\\n\\n(cid:1)\\n\\nThe difficulty is to produce enough cohomology classes in S∗\\n\\nL to show that the right side of the above inequality is small. Following Kolyvagin, an Euler system is S∗ L for a large (infinite) collection of sets of a compatible collection of classes κ( )) primes is related to resℓ(κ( )). Once an Euler system is given, Kolyvagin has an inductive procedure for choosing a set\\n\\n) L\\n\\n∈\\n\\n. Loosely speaking, compatible means that if ℓ /\\n\\n, then resℓ(κ(\\n\\nℓ\\n\\nL\\n\\n∈ L\\n\\nL ∪ {\\n\\n}\\n\\nL\\n\\nsuch that\\n\\nL\\n\\nresL is injective on S(M ),\\n\\n•\\n\\nP⊆L ker(θκ(P)) can be computed in terms of κ( ∅\\n\\n).\\n\\nT\\n\\n18\\n\\nK. RUBIN AND A. SILVERBERG\\n\\nS∗\\n\\nS∗\\n\\n, then S∗\\n\\nL.)\\n\\nL, so κ(\\n\\n)\\n\\n(Note that if\\n\\nP ⊆\\n\\nP For several important Selmer groups it is possible to construct Euler systems for\\n\\n∈\\n\\nP ⊆ L\\n\\nwhich Kolyvagin’s procedure produces a set\\n\\nactually giving an equality\\n\\nL ker(θκ(P))\\n\\n#(S(M )) = #\\n\\n.\\n\\n(cid:0) \\\\P⊆L This is what Wiles needs to do for the Selmer group SD(Vn). There are several examples in the literature where this kind of argument is worked out in some detail. For the simplest case, where the Selmer group in question is the ideal class group ) are constructed from cyclotomic units, of a real abelian number field and the κ( L see [29]. For other cases involving ideal class groups and Selmer groups of elliptic curves, see [19], [31], [30], [13].\\n\\n(cid:1)\\n\\n5.4. Wiles’ geometric Euler system. The task now is to construct an Euler system of cohomology classes with which to bound #(SD(Vn)) using Kolyvagin’s method. This is the most technically difficult part of Wiles’ proof and is the part of Wiles’ work he referred to as not yet complete in his December announcement. We give only general remarks about Wiles’ construction.\\n\\nThe first step in the construction is due to Flach [10]. He constructed classes consisting of just one prime. This allows one to bound the ) L\\n\\nS∗\\n\\nκ( exponent of SD(Vn), but not its order.\\n\\nL for sets\\n\\n∈\\n\\nL\\n\\nEvery Euler system starts with some explicit, concrete objects. Earlier examples of Euler systems come from cyclotomic or elliptic units, Gauss sums, or Heegner points on elliptic curves. Wiles (following Flach) constructs his cohomology classes from modular units, i.e., meromorphic functions on modular curves which are holo- morphic and nonzero away from the cusps. More precisely, κ( ) comes from an explicit function on the modular curve X1(L, N ), the curve obtained by taking the quotient space of the upper half plane by the action of the group\\n\\nL\\n\\na b c d\\n\\nSL2(Z) : c\\n\\n1 (mod L) } ≡ ℓ∈L ℓ and where N is the N of (T2) of\\n\\n0\\n\\n(mod LN ),\\n\\na\\n\\nd\\n\\n,\\n\\nΓ1(L, N ) =\\n\\n∈\\n\\n≡\\n\\n≡\\n\\n{ (cid:1) (cid:0) and adjoining the cusps, where L = The construction and study of the classes κ( [8], [9] and others.\\n\\n4.2. ) rely heavily on results of Faltings\\n\\n§\\n\\nL\\n\\nQ\\n\\n5.5. Wiles’ results. Wiles announced two main results (Theorems 5.3 and 5.4 below) in the direction of Mazur’s conjecture, under two different sets of hypotheses on the representation ¯ρ. Theorem 5.3 implies the Semistable Taniyama-Shimura Conjecture and Fermat’s Last Theorem. Wiles’ proof of Theorem 5.3 depends on the not-yet-complete construction of an appropriate Euler system (as in 5.4), while his proof of Theorem 5.4 (though not yet fully checked) does not. For Theorem 5.4, Wiles bounds the Selmer group of 5.2 without constructing a new Euler system, by using results from the Iwasawa theory of imaginary quadratic fields. (These results in turn rely on Kolyvagin’s method and the Euler system of elliptic units; see [31].)\\n\\n§\\n\\n§\\n\\nSince for ease of exposition we defined modularity of representations in terms of Γ0(N ) instead of Γ1(N ), the theorems stated below are weaker than those an- nounced by Wiles, but have the same applications to elliptic curves. (Note that by our definition of type-\\n\\n, if ¯ρ is type-\\n\\n, then det(¯ρ) = ¯εp.)\\n\\nD\\n\\nD\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n19\\n\\nIf ¯ρ is a representation of GQ on a vector space V , Sym2(¯ρ) denotes the repre-\\n\\nsentation on the symmetric square of V induced by ¯ρ.\\n\\nTheorem 5.3 (Wiles). Suppose p, k, the following additional conditions :\\n\\n, ¯ρ, and\\n\\nare as in\\n\\n4.2 and ¯ρ satisfies\\n\\nD\\n\\nO\\n\\n§\\n\\n(i) Sym2(¯ρ) is absolutely irreducible, (ii) if ¯ρ is ramified at q and q (iii) if p is 3 or 5, then for some prime q, p divides #(¯ρ(Iq)).\\n\\n= p, then the restriction of ¯ρ to Dq is reducible,\\n\\n6\\n\\nT is an isomorphism.\\n\\nThen ϕ : R\\n\\n→\\n\\nSince Theorem 5.3 does not yield the full Mazur Conjecture (Conjecture 4.2) for 2 to see which elliptic curves §\\n\\np = 3 and 5, we need to reexamine the arguments of E can be proved modular using Theorem 5.3 applied to ¯ρE,3 and ¯ρE,5.\\n\\nHypothesis (i) of Theorem 5.3 will be satisfied if the image of ¯ρE,p is sufficiently large in GL2(Fp) (for example, if ¯ρE,p is surjective). For p = 3 and p = 5, if ¯ρE,p satisfies hypothesis (iii) and is irreducible, then it satisfies hypothesis (i).\\n\\nIf E is semistable, p is an odd prime, and ¯ρE,p is irreducible and modular, then (see the proof of Proposition 3.3) and ¯ρE,p satisfies (ii) ¯ρE,p is D 14 of Appendix C of [39]). Therefore by Propositions and (iii) (use Tate curves; see 4.3 and 3.3, Theorem 5.3 implies that the Semistable Modular Lifting Conjecture (Conjecture 2.1) holds for p = 3 and for p = 5. As shown in 2, the Semistable Taniyama-Shimura Conjecture and Fermat’s Last Theorem follow.\\n\\nmodular for some\\n\\nD\\n\\n§\\n\\n§\\n\\nTheorem 5.4 (Wiles). Suppose p, k, contains no nontrivial p-th roots of unity. Suppose also that there are an imaginary quadratic field F of discriminant prime to p and a character χ : Gal( ¯Q/F ) × such that T is the induced representation Indχ of GQ is a ( an isomorphism.\\n\\n, ¯ρ, and\\n\\nare as in\\n\\n4.2 and\\n\\nD\\n\\nO\\n\\n§\\n\\nO\\n\\n→ O\\n\\n)-lifting of ¯ρ. Then ϕ : R\\n\\n,\\n\\nD\\n\\nO\\n\\n→\\n\\nCorollary 5.5 (Wiles). Suppose E is an elliptic curve over Q with complex mul- tiplication by an imaginary quadratic field F and p is an odd prime at which E has good reduction. If E′ is an elliptic curve over Q satisfying\\n\\nE′ has good reduction at p and ¯ρE′,p is isomorphic to ¯ρE,p,\\n\\n•\\n\\nthen E′ is modular.\\n\\nProof of corollary. Let p be a prime of F containing p, and define = the ring of integers of the completion of F at p,\\n\\nO • • •\\n\\n/p primes at which E or E′ has bad reduction\\n\\nk = Σ = t = ordinary if E has ordinary reduction at p, t = flat if E has supersingular reduction at p,\\n\\n,\\n\\nO {\\n\\nO\\n\\np\\n\\n,\\n\\n} ∪ {\\n\\n}\\n\\n= (Σ, t).\\n\\nD\\n\\nLet\\n\\nχ : Gal( ¯Q/F )\\n\\nAutO(E[p∞]) ∼=\\n\\n×\\n\\n→\\n\\nO\\n\\nbe the character giving the action of Gal( ¯Q/F ) on E[p∞] (where E[p∞] is the group of points of E killed by the endomorphisms of E which lie in some power of p). It is not hard to see that ρE,p ⊗ O\\n\\nis isomorphic to Indχ.\\n\\n20\\n\\nK. RUBIN AND A. SILVERBERG\\n\\nSince E has complex multiplication, it is well known that E and ¯ρE,p are mod- ular. Since E has good reduction at p, it can be shown that the discriminant of contains no nontrivial p-th roots of unity. One can show F is prime to p and that all of the hypotheses of Theorem 5.4 are satisfied with ¯ρ = ¯ρE,p ⊗ k. By our assumptions on E′, ρE′,p ⊗ O )-lifting of ¯ρ, and we conclude (using the D same reasoning as in the proofs of Propositions 3.3 and 4.3) that ρE′,p is modular and hence E′ is modular.\\n\\nO\\n\\nis a (\\n\\n,\\n\\nO\\n\\nRemarks. (i) The elliptic curves E′ of Corollary 5.5 are not semistable.\\n\\n(ii) Suppose E and p are as in Corollary 5.5 and p = 3 or 5. As in Appendix B.2 one can show that the elliptic curves E′ over Q with good reduction at p and with ¯ρE′,p isomorphic to ¯ρE,p give infinitely many C-isomorphism classes.\\n\\nExample. Take E to be the elliptic curve defined by\\n\\ny2 = x3\\n\\nx2\\n\\n3x\\n\\n1.\\n\\n−\\n\\n−\\n\\n−\\n\\nThen E has complex multiplication by Q(√ Define polynomials\\n\\n2), and E has good reduction at 3.\\n\\n−\\n\\n1512t3 3, a4(t) = a6(t) = 40824t6 + 31104t5 + 8370t4 + 504t3\\n\\n2430t4\\n\\n396t2\\n\\n56t\\n\\n−\\n\\n−\\n\\n−\\n\\n−\\n\\n−\\n\\n148t2\\n\\n24t\\n\\n1,\\n\\n−\\n\\n−\\n\\n−\\n\\nQ let Et be the elliptic curve\\n\\nand for each t\\n\\n∈\\n\\ny2 = x3\\n\\nx2 + a4(t)x + a6(t)\\n\\n−\\n\\nQ, ¯ρEt,3 is isomorphic to (note that E0 = E). It can be shown that for every t 0 or 1 (mod 3) (or more generally if t = 3a/b or t = 3a/b + 1 ¯ρE,3. If t with a and b integers and b not divisible by 3), then Et has good reduction at 3, for instance because the discriminant of Et is\\n\\n∈\\n\\nZ and t\\n\\n∈\\n\\n≡\\n\\n29(27t2 + 10t + 1)3(27t2 + 18t + 1)3.\\n\\nThus for these values of t, Corollary 5.5 shows that Et is modular and so is any elliptic curve over Q isomorphic over C to Et, i.e., any elliptic curve over Q with j-invariant equal to\\n\\n3\\n\\n4(27t2 + 6t + 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + 1)\\n\\n.\\n\\n(cid:18)\\n\\n(cid:19)\\n\\nThis explicitly gives infinitely many modular elliptic curves over Q which are\\n\\nnonisomorphic over C.\\n\\n(For definitions of complex multiplication, discriminant, and j-invariant, see any\\n\\nstandard reference on elliptic curves, such as [39].)\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n21\\n\\nAppendix A. Galois groups and Frobenius elements\\n\\nWrite GQ = Gal( ¯Q/Q). If q is a prime number and\\n\\nis a prime ideal dividing\\n\\nQ\\n\\nq in the ring of integers of ¯Q, there is a filtration\\n\\nGQ\\n\\nDQ ⊃\\n\\nIQ\\n\\n⊃ where the decomposition group DQ and the inertia group IQ are defined by\\n\\nDQ = IQ =\\n\\nσ\\n\\nGQ : σ ∈ Q ∈ DQ : σx\\n\\n=\\n\\n,\\n\\n{\\n\\nQ} x (mod\\n\\nσ\\n\\n) for all algebraic integers x }\\n\\n.\\n\\n≡ { There are natural identifications\\n\\nQ\\n\\nDQ/IQ ∼= Gal( ¯Fq/Fq),\\n\\nDQ ∼= Gal( ¯Qq/Qq),\\n\\nxq of GQ\\n\\nand FrobQ ∈ Gal( ¯Fq/Fq). If and\\n\\nDQ/IQ denotes the inverse image of the canonical generator x\\n\\n7→ for some σ\\n\\n′ is another prime ideal above q, then\\n\\n′ = σ\\n\\nQ DQ′ = σDQσ−1,\\n\\nQ\\n\\nQ\\n\\n∈\\n\\nFrobQ′ = σFrobQσ−1.\\n\\nIQ′ = σIQσ−1,\\n\\nSince we will care about these objects only up to conjugation, we will write Dq and GQ for any representative of a FrobQ. If ρ is a represen- Iq. We will write Frobq ∈ tation of GQ which is unramified at q, then trace(ρ(Frobq)) and det(ρ(Frobq)) are well defined independent of any choices.\\n\\nAppendix B. Some details on the proof of Proposition 2.4\\n\\nB.1. The modular curve X0(15) can be viewed as a curve defined over Q in such a way that the noncusp rational points correspond to isomorphism classes (over C) E( ¯Q) is a subgroup of pairs (E′, 42), of order 15 stable under GQ. An equation for X0(15) is y2 = x(x + 32)(x the elliptic curve discussed in 1. There are eight rational points on X0(15), four of § which are cusps. There are four modular elliptic curves, corresponding to a modular form for Γ0(50) (see p. 86 of [1]), which lie in the four distinct C-isomorphism classes that correspond to the noncusp rational points on X0(15).\\n\\n) where E′ is an elliptic curve over Q and\\n\\nC\\n\\nC ⊂\\n\\n−\\n\\nTherefore every elliptic curve over Q with a GQ-stable subgroup of order 15 is modular. Equivalently, if E is an elliptic curve over Q and both ¯ρE,3 and ¯ρE,5 are reducible, then E is modular.\\n\\nB.2. Fix a semistable elliptic curve E over Q. We will show that there are infinitely many semistable elliptic curves E′ over Q such that\\n\\n(i) ¯ρE′,5 is isomorphic to ¯ρE,5, and (ii) ¯ρE′,3 is irreducible. Let\\n\\n1 0 0 1\\n\\na b c d\\n\\na b c d\\n\\nSL2(Z) :\\n\\n(mod 5) }\\n\\n.\\n\\nΓ(5) =\\n\\n≡\\n\\n∈\\n\\n{\\n\\nLet X be the twist of the classical modular curve X(5) (see [35]) by the cocycle (cid:0) induced by ¯ρE,5, and let S be the set of cusps of X. Then X is a curve defined over Q which has the following properties. The rational points on X − (E′, φ) where E′ is an elliptic curve over Q and φ : E[5] module isomorphism.\\n\\n(cid:1)\\n\\n(cid:0)\\n\\n(cid:1)\\n\\n(cid:1)\\n\\n(cid:0)\\n\\nS correspond to isomorphism classes of pairs E′[5] is a GQ-\\n\\n\\n\\n→\\n\\n22\\n\\nK. RUBIN AND A. SILVERBERG\\n\\nS is four copies of H/Γ(5), so each component of\\n\\nAs a complex manifold X X has genus zero.\\n\\n\\n\\n−\\n\\nLet X 0 be the component of X containing the rational point corresponding to (E, identity). Then X 0 is a curve of genus zero defined over Q with a rational point, so it has infinitely many rational points. We want to show that infinitely many of these points correspond to semistable elliptic curves E′ with ¯ρE′,3 irreducible.\\n\\nThere is another modular curve ˆX defined over Q, with a finite set ˆS of cusps,\\n\\nwhich has the following properties. The rational points on ˆX (E′, φ, module isomorphism, and As a complex manifold ˆX The map that forgets the subgroup X defined over Q and of degree [Γ(5) : Γ(5)\\n\\nˆS correspond to isomorphism classes of triples E′[5] is a GQ-\\n\\n\\n\\n−\\n\\n) where E′ is an elliptic curve over Q, φ : E[5]\\n\\nC\\n\\n→\\n\\nE′[3] is a GQ-stable subgroup of order 3.\\n\\nC ⊂ −\\n\\nˆS is four copies of H/(Γ(5)\\n\\nΓ0(3)).\\n\\n•\\n\\n∩ induces a surjective morphism θ : ˆX\\n\\nC\\n\\n→\\n\\nΓ0(3)] = 4.\\n\\n∩\\n\\nLet ˆX 0 be the component of ˆX which maps to X 0. The function field of X 0 is Q(t), and the function field of ˆX 0 is Q(t)[x]/f (t, x) where f (t, x) Q(t)[x] is irreducible and has degree 4 in x. If t′ Q is sufficiently close 5-adically to the value of t which corresponds to E, then the corresponding elliptic curve is semistable at Q so that f (t1, x) is 5. By the Hilbert Irreducibility Theorem we can find a t1 ∈ irreducible in Q[x]. It is possible to fix a prime ℓ = 5 such that f (t1, x) has no roots modulo ℓ. If t′ Q is sufficiently close ℓ-adically to t1, then f (t′, x) has no rational roots, and thus t′ corresponds to a rational point of X 0 which is not the image of a rational point of ˆX 0. Therefore there are infinitely many elliptic curves E′ over Q which are semistable at 5 and satisfy\\n\\n∈\\n\\n∈\\n\\n6\\n\\n∈\\n\\n(i) E′[5] ∼= E[5] as GQ-modules, and (ii) E′[3] has no subgroup of order 3 stable under GQ.\\n\\nIt follows from (i) and the semistability of E that E′ is semistable at all primes = 5, and thus E′ is semistable. We therefore have infinitely many semistable q elliptic curves E′ which satisfy the desired conditions.\\n\\n6\\n\\nAppendix C. Representation types\\n\\nSuppose A is a complete noetherian local Zp-algebra and ρ : GQ\\n\\nGL2(A) is a |Dp for the restriction of ρ to the decomposition group Dp.\\n\\n→\\n\\nrepresentation. Write ρ We say ρ is\\n\\nordinary at p if ρ\\n\\n|Dp is (after a change of basis, if necessary) of the form flat at p if ρ is not ordinary, and for every ideal a of finite index in A, the (cid:0) |Dp modulo a is the representation associated to the ¯Qp-points reduction of ρ of a finite flat group scheme over Zp.\\n\\n\\n\\n∗ ∗ 0 χ\\n\\nwhere χ is unramified and the * are functions from Dp to A;\\n\\n(cid:1)\\n\\n\\n\\nAppendix D. Selmer groups\\n\\nWith notation as in\\n\\n5 (see especially §\\n\\n5.2), define\\n\\n§\\n\\n[ǫ]/(ǫ2, mn)\\n\\nOn =\\n\\nO\\n\\nA REPORT ON WILES’ CAMBRIDGE LECTURES\\n\\n23\\n\\nwhere ǫ is an indeterminate. Then v\\n\\n1 + ǫv defines an isomorphism\\n\\n7→ On) : δ GL2(\\n\\n∼ ∈ → { HomO(pR/p2 R,\\n\\n(16)\\n\\n1 (mod ǫ) } /mn) there is a unique -algebra homomorphism → On whose restriction to pR is ǫα. Composing with the representation ρR On. (In particular ρ0 )-lifting obtained when α = 0.) Define a one-cocycle cα on GQ\\n\\nδ\\n\\n.\\n\\nVn\\n\\n≡\\n\\nFor every α\\n\\nO\\n\\nO\\n\\n∈\\n\\nψα : R of Theorem 4.1 gives a ( denotes the ( by\\n\\n,\\n\\n)-lifting ρα = ψα ◦\\n\\nρR of ¯ρ to\\n\\nD\\n\\nO\\n\\n,\\n\\nD\\n\\nO\\n\\ncα(g) = ρα(g)ρ0(g)−1.\\n\\nH 1(Q, Vn). This defines a\\n\\nSince ρα ≡ homomorphism\\n\\nρ0 (mod ǫ), using (16) we can view cα ∈\\n\\ns : HomO(pR/p2 R,\\n\\n/mn)\\n\\nH 1(Q, Vn),\\n\\nO and it is not difficult to see that s is injective. The fact that ρ0 and ρα are type- D gives information about the restrictions resq(cα) for various primes q, and using this H 1(Q, Vn) and verifies that s information Wiles defines a Selmer group SD(Vn) is an isomorphism onto SD(Vn).\\n\\n→\\n\\n⊂\\n\\nReferences\\n\\n[1] B. Birch and W. Kuyk, eds., Modular functions of one variable. IV, Lecture Notes in Math.,\\n\\nvol. 476, Springer-Verlag, New York, 1975, pp. 74–144.\\n\\n[2] J. Buhler, R. Crandall, R. Ernvall, and T. 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Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc.\\n\\n(N.S.) 5 (1981), 173–175.\\n\\n[42] A. Weil, ¨Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math.\\n\\nAnn. 168 (1967), 149–156.\\n\\nDepartment of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: rubin@math.ohio-state.edu\\n\\nDepartment of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: silver@math.ohio-state.edu' metadata={'source': '/var/folders/l1/lphj87z16c3282pjwy91wtm80000gn/T/tmpdh5kk5yb/tmp.pdf'}\n",
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