## Galois representations and modular forms (1995)

Venue: | Bull. Amer. Math. Soc |

Citations: | 11 - 0 self |

### BibTeX

@ARTICLE{Ribet95galoisrepresentations,

author = {Kenneth A. Ribet},

title = {Galois representations and modular forms},

journal = {Bull. Amer. Math. Soc},

year = {1995},

volume = {32},

pages = {375--402}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. In this article, I discuss material which is related to the recent

### Citations

390 | Modular elliptic curves and Fermat’s Last Theorem
- Wiles
(Show Context)
Citation Context ...eorem of an article written jointly by Richard Taylor and Andrew Wiles [87]; these authors announced their result at the same time that Wiles disseminated a revised version of his original manuscript =-=[90]-=-. It is premature to undertake a detailed analysis of the results of Wiles and TaylorWiles. The aim of this survey is more modest: to present an introduction to the Received by the editors November 22... |

382 |
Commutative Ring Theory
- Matsumura
- 1986
(Show Context)
Citation Context ... 12. Gorenstein and complete intersection conditions The conjecture that ϕΣ is an isomorphism is proved in [87, 90]. The proof relies on standard notions of commutative algebra which are discussed in =-=[43]-=- and in [41].s22 KENNETH A. RIBET The argument of Wiles and Taylor-Wiles proceeds from the definition of TΣ as a completion of the ring generated by the Hecke operators Tn acting on a specific space o... |

239 | Ring theoretic properties of certain Hecke algebras
- Taylor, Wiles
(Show Context)
Citation Context ...llow from a proof that certain Hecke algebras are complete intersection rings. This statement about Hecke algebras is the main theorem of an article written jointly by Richard Taylor and Andrew Wiles =-=[87]-=-; these authors announced their result at the same time that Wiles disseminated a revised version of his original manuscript [90]. It is premature to undertake a detailed analysis of the results of Wi... |

238 |
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern
- Faltings
- 1983
(Show Context)
Citation Context ... of G. Faltings [19, §5] states that, conversely, two elliptic curves A and A ′ over Q are isogenous if #( Ãp(Z/pZ)) = #( Ã′ p(Z/pZ)) for all primes p at which the two curves have good reduction. (In =-=[19]-=-, Faltings proves a more general statement about homomorphisms between abelian varieties over number fields, thus confirming conjectures of J. Tate.) Each elliptic curve A over Q has a conductor, whic... |

152 | Modular curves and the Eisenstein ideal - Mazur - 1977 |

152 | The arithmetic of elliptic curves, Graduate Texts - Silverman |

146 |
Good reduction of abelian varieties
- Serre, Tate
- 1968
(Show Context)
Citation Context ...ct an equivalence relation.) It is not difficult to show that two isogenous elliptic curves over Q have the same primes of bad reduction. (This follows from the criterion of Néron-Ogg-Shafarevich see =-=[76]-=- or [83, Ch. VII, §7].) Moreover, if A and A ′ are isogenous over Q , then for each good reduction prime p , one has #(Ãp(Z/pZ)) = #(Ã ′ p (Z/pZ)). In other words, the numbers bp are the same for A an... |

97 |
les représentations modulaires de degré 2 de Gal
- Serre, Sur
(Show Context)
Citation Context ...ves over the complex field C (see [69]). In my lecture at the Mathfest, I stressed this achievement of Wiles and discussed the analogy between the Taniyama-Shimura conjecture and conjectures of Serre =-=[74]-=- about two-dimensional Galois representations. As I began to revise these notes, I found that the situation had changed dramatically for the better. Wiles announced in October 1994 that the bound he s... |

92 |
Elliptic curves
- Knapp
- 1992
(Show Context)
Citation Context ...lar forms. Among reference books in the subject, one might cite [36], [53], and [79]. Also, several books on elliptic curves contain substantial material on modular forms; in particular, Knapp’s book =-=[35]-=- has been recommended to the author as a good source for an overview of the Eichler-Shimura theory relating elliptic curves to certain modular forms. For a first introduction to modular forms, a fine ... |

78 |
algorithm for determining the type of a singular fiber in an elliptic pencil
- Tate
- 1975
(Show Context)
Citation Context ...arity is detected by the non-vanishing of the discriminant of the equation. (The discriminant is a polynomial in the αi which will not be reproduced here; it can be found, for instance, on page 36 of =-=[85]-=-.) A short calculation shows that the plane curve with the indicated equation has exactly one point which does not lie on the affine plane; we take this point to be the distinguished point O . Using t... |

75 |
Introduction to modular forms
- Lang
- 1976
(Show Context)
Citation Context ...ey with a disproportionately large list of references. 2. Modular forms We begin by summarizing some background material concerning modular forms. Among reference books in the subject, one might cite =-=[36]-=-, [53], and [79]. Also, several books on elliptic curves contain substantial material on modular forms; in particular, Knapp’s book [35] has been recommended to the author as a good source for an over... |

73 |
Hecke operators on Γ0(m
- Atkin, Lehner
- 1970
(Show Context)
Citation Context ...r words, the commuting operators Tn are not necessarily all diagonalizable. This problem, which arises from those n which have a common factor with N , can be repaired by the introduction of newforms =-=[2]-=-. Briefly, a newform is a normalized eigenform f = � anqn for which the space � � � � g ∈ S(N) � g|Tn = ang for all n prime to N contains only f and its multiples. Atkin and Lehner showed in 1970 that... |

70 |
Formes modulaires de poids 1
- Deligne, Serre
- 1974
(Show Context)
Citation Context ...and k>1 are quite distinct in flavor. In the former case, the representations ρf ,λ may be viewed as the set of reductions of a single continuous representation ρf with finite image. For details, see =-=[16]-=-. 7. Serre’s conjectures We shall give a brief summary of conjectures made by J.-P. Serre in [74]. A recent article by H. Darmon [15] discusses the conjectures more extensively and emphasizes applicat... |

70 |
change for GL(2
- Langlands
- 1980
(Show Context)
Citation Context ...ork with a single prime ℓ . The approach of [90] capitalizes on the fact that the finite groups GL(2 , F2) andGL(2 , F3) are solvable. This circumstance enables one to apply deep results of Langlands =-=[40]-=- and Tunnell [88] to prove that ρA,ℓ is modular for ℓ ≤ 3 , cf. [68, §2.3]. (The relevant results of Langlands are those concerning the theory of base change à la Saito-Shintani. For expositions of th... |

56 |
Galois representations attached to eigenforms with nebentypus, In: Modular functions of one variable V
- Ribet
- 1977
(Show Context)
Citation Context ...is odd. This parity statement generalizes to modular forms “with Nebentypus” whose weights are not necessarily two. Here is a quick synopsis of the situation; some relevant references are provided in =-=[60]-=-. For integers k ≥ 1, N ≥1 and characters ɛ:(Z/NZ) ∗ → C ∗ , one considers the space Sk(N ,ɛ)ofweight-kcusp forms with character ɛ on Γ0(N); wehave S(N)=S2(N,1) . This space is automatically zero unle... |

48 |
les représentations modulaires de degré 2 de
- Serre, Sur
- 1987
(Show Context)
Citation Context ...ves over the complex field C (see [69]). In my lecture at the Mathfest, I stressed this achievement of Wiles and discussed the analogy between the Taniyama-Shimura conjecture and conjectures of Serre =-=[75]-=- about two-dimensional Galois representations. As I began to revise these notes, I found that the situation had changed dramatically for the better. Wiles announced in October, 1994 that the bound he ... |

47 | On the Counting Functions
- Li
- 1993
(Show Context)
Citation Context ... new S(M) � . M|N d| N M We conclude this discussion with the remark that the work of Atkin and Lehner was generalized by T. Miyake [52] in the setting of automorphic forms on GL(2) and then by W. Li =-=[42]-=- in the setting of classical modular forms. 3. Elliptic curves We next introduce some foundational concepts pertaining to elliptic curves. For an in-depth treatment of these concepts, the reader may c... |

45 |
Il n’y a pas de variété abélienne sur Z
- Fontaine
- 1985
(Show Context)
Citation Context ..., potentially good reduction. Hence if it is semistable, it has good reduction everywhere. However, a theorem of Tate states that there is no elliptic curve over Q with everywhere good reduction, cf. =-=[21]-=-.) Accordingly, the theorem of [81] cannot be used to prove Fermat’s Last Theorem. A key piece of information about the extension Kn/Q (which depends on A as well as on n) is that its discriminant is ... |

44 | locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Séminaire Delange-Pisot-Poitou 1969/70 - Serre |

42 |
les représentations ℓ-adiques associees aux formes modulaires de
- Carayol
- 1986
(Show Context)
Citation Context ...e fact that f is necessarily built from anewformin S(M) for some unique divisor M of N . It is then true that the conductor of A is precisely this divisor M . This theorem was proved by H. Carayol in =-=[9]-=-, following work of Shimura, Igusa, Deligne and Langlands. A formulation of the Taniyama-Shimura conjecture with a completely different flavor is provided by Mazur’s article [48]. In this article, Maz... |

36 |
A finiteness theorem for the symmetric square of an elliptic curve
- Flach
- 1992
(Show Context)
Citation Context ... of Galois representations by working directly with ϕΣ .Despite conjectures by Mazur [51, p. 85] and Gouvêa [27, p. 108] to the effect that ϕΣ is an isomorphism in the ordinary case, and Theorem 2 of =-=[20]-=-, I was not prepared for the revelation that ϕΣ could be studied fruitfully. 12. Gorenstein and complete intersection conditions The conjecture that ϕΣ is an isomorphism is proved in [87, 90]. The pro... |

36 |
Geometric Galois representations
- Fontaine, Mazur
- 1993
(Show Context)
Citation Context ...ma-Shimura conjecture can be generalized still further. Indeed, as10 KENNETH A. RIBET common generalization of Conjectures 1 and 2 was presented in the 1995 Berkeley Ph.D. thesis of E. Pyle. See also =-=[22]-=-. 5. Galois representations attached to elliptic curves Let A be an elliptic curve over Q .AmodelforA(K)whenK=Cis given by the Weierstraß theory of complex analysis: the group A(C) is the complex toru... |

30 | Artin’s conjecture for representations of octahedral type - Tunnell - 1981 |

26 |
On a variation of Mazur’s deformation functor
- Ramakrishna
- 1993
(Show Context)
Citation Context ...ons is due to Mazur [47]. As F. Gouvêa reports in his recent survey [29], the deformation viewpoint has evolved considerably since [47] first appeared. In particular, a 1993 article by R. Ramakrishna =-=[59]-=- introduces foundational tools which Wiles requires in [90]; Ramakrishna shows that deformation theory can be applied to study the family of lifts which satisfy the local conditions to which we have b... |

23 |
Kuyk (eds.), Modular functions of one variable
- Birch, W
- 1975
(Show Context)
Citation Context ...order 5 whenever A has good reduction at p . Accordingly, the number #( Ãp(Z/pZ)) is divisible by 5, so that bp ≡ p+1 mod 5 for all primes p which do not divide the discriminant of A . Now Table 1 of =-=[4]-=- informs us that the minimal discriminant of A is −11 . Therefore, integers bp are defined for all p �=11.When p is small, it is not hard to compute bp ; one finds b2 = −2, b3 = −1, b5 =1,andsoon. (Th... |

22 | Représentations galoisiennes, différentielles de Kähler et “conjectures principales - Mazur, Tilouine - 1990 |

21 |
Links between stable elliptic curves and certain Diophantine equations
- Frey
- 1986
(Show Context)
Citation Context ...e Fermat’s Last Theorem outright or else to link it to established or emerging conjectures such as the abc conjecture and Szpiro’s conjecture. These latter conjectures are treated by such articles as =-=[23, 24, 32, 39, 50]-=- and [56]. From the point of view of Szpiro’s conjecture and the abc conjecture, the surprising feature of Frey’s curve is the size of its discriminant rather than any special properties of the discri... |

19 |
Silverman (Editors) – Arithmetic geometry
- Cornell, H
- 1986
(Show Context)
Citation Context ...n connection with Fermat’s Last Theorem and in other applications. This is a good point to insert a short digression about abelian varieties, whose arithmetic theory is exposed in various chapters of =-=[12]-=- and the recent graduate-level text [54]; see also the books by such authors as A. Weil, S. Lang [37], D. Mumford, H. P. F. Swinnerton-Dyer, G. Kempf and H. Lange-Ch. Birkenhake. As was mentioned very... |

19 |
Theory as Gadfly
- “Number
- 1991
(Show Context)
Citation Context ...roved by H. Carayol in [9], following work of Shimura, Igusa, Deligne and Langlands. A formulation of the Taniyama-Shimura conjecture with a completely different flavor is provided by Mazur’s article =-=[48]-=-. In this article, Mazur rephrases the conjecture as a statement about the Riemann surface associated with an elliptic curve over Q .If Ais such an elliptic curve, we write A(C) for this Riemann surfa... |

16 |
Elliptic curves and wild ramification
- Ogg
- 1967
(Show Context)
Citation Context ...of an elliptic curve tends to be divisible only by low powers of its prime divisors. (In particular, a prime p ≥ 5 can divide the conductor at most to the second power.) A beautiful formula of A. Ogg =-=[57]-=- expresses the conductor of a given curve A in terms of the minimal discriminant and the Néron model of A . (The latter is a curve over Z which may be regarded as the “best possible” modelsGALOIS REPR... |

13 |
and functional equations
- Li
- 1975
(Show Context)
Citation Context ...d| N M ( new δd S(M) ) . We conclude this discussion with the remark that the work of Atkin and Lehner was generalized by T. Miyake [52] in the setting of automorphic forms on GL(2) and then by W. Li =-=[42]-=- in the setting of classical modular forms. 3. Elliptic curves We next introduce some foundational concepts pertaining to elliptic curves. For an in-depth treatment of these concepts, the reader may c... |

12 |
The refined conjecture of Serre
- Diamond
- 1995
(Show Context)
Citation Context ...njecture, Fermat’s Last Theorem, and a host of other assertions! It has gradually emerged that the qualitative statement and its strong cousin are in fact equivalent, at least when ℓ>2 ; see [67] and =-=[17]-=- for a proof of the equivalence. Hence it is now possible to use the singular term “Serre’ss16 KENNETH A. RIBET conjecture” to refer to what was initially a package of interrelated conjectures. Perhap... |

12 |
Nouvelles approches du “théorème” de Fermat, Séminaire Bourbaki no
- Oesterlé
- 1988
(Show Context)
Citation Context ...utright or else to link it to established or emerging conjectures such as the abc conjecture and Szpiro’s conjecture. These latter conjectures are treated by such articles as [23, 24, 32, 39, 50] and =-=[56]-=-. From the point of view of Szpiro’s conjecture and the abc conjecture, the surprising feature of Frey’s curve is the size of its discriminant rather than any special properties of the discriminant’s ... |

11 |
The solution of a base change problem for GL(2) (following Langlands
- Gérardin, Labesse
- 1979
(Show Context)
Citation Context ... that ρA,ℓ is modular for ℓ ≤ 3 , cf. [68, §2.3]. (The relevant results of Langlands are those concerning the theory of base change à la Saito-Shintani. For expositions of these results, see [25] and =-=[26]-=-.) Wiles’s basic idea is to prove that if ℓ is a prime for which ρA,ℓ is modular, then ρA,ℓ∞ is automatically modular (and hence A is a modular elliptic curve). In thinking about the jump from ρA,ℓ to... |

10 |
Introduction to Fermat’s last theorem
- Cox
- 1994
(Show Context)
Citation Context ...with Wiles’s 1993 announcement. Among these are the author’s news item in the Notices [65] and his article with Brian Hayes in American Scientist [30], two pieces in the American Mathematical Monthly =-=[13, 28]-=-, the report by K. Rubin and A. Silverberg in this Bulletin [68], a long survey by H. Darmon [14], an article for undergraduates and their teachers by N. Boston [6], and an elementary article by A. As... |

9 |
Jr.: Complete intersections and Gorenstein rings
- Lenstra
- 1995
(Show Context)
Citation Context ...ein and complete intersection conditions The conjecture that ϕΣ is an isomorphism is proved in [87, 90]. The proof relies on standard notions of commutative algebra which are discussed in [43] and in =-=[41]-=-.s22 KENNETH A. RIBET The argument of Wiles and Taylor-Wiles proceeds from the definition of TΣ as a completion of the ring generated by the Hecke operators Tn acting on a specific space of classical ... |

9 |
On automorphic forms on GL2 and Hecke operators
- Miyake
- 1971
(Show Context)
Citation Context ...e the subspace of S(M) spanned by its newforms. Then S(N) = ⊕ ⊕ M|N d| N M ( new δd S(M) ) . We conclude this discussion with the remark that the work of Atkin and Lehner was generalized by T. Miyake =-=[52]-=- in the setting of automorphic forms on GL(2) and then by W. Li [42] in the setting of classical modular forms. 3. Elliptic curves We next introduce some foundational concepts pertaining to elliptic c... |

8 |
Introduction to Abelian varieties
- Murty
- 1993
(Show Context)
Citation Context ...and in other applications. This is a good point to insert a short digression about abelian varieties, whose arithmetic theory is exposed in various chapters of [12] and the recent graduate-level text =-=[54]-=-; see also the books by such authors as A. Weil, S. Lang [37], D. Mumford, H. P. F. Swinnerton-Dyer, G. Kempf and H. Lange-Ch. Birkenhake. As was mentioned very briefly above, an abelian variety over ... |

8 |
topics in the arithmetic of elliptic curves, Graduate Texts
- Advanced
- 1994
(Show Context)
Citation Context ...review written by W. R. Hearst III and the author [31]. One book which has appeared since that review was written is the recent Advanced topics in the arithmetic of elliptic curves by J. H. Silverman =-=[84]-=-. Elliptic curves are distinguished by the fact that they are simultaneously curves, i.e., varieties of dimension 1, and abelian varieties, projective varieties which are endowed with a group law. The... |

8 | Une interprétation des congruences relatives à la fonction τ de Ramanujan, Séminaire Delange–Pisot–Poitou - Serre - 1969 |

6 |
Complete systems of two addition laws for elliptic curves
- Bosma, Lenstra
(Show Context)
Citation Context ...ly of polynomial equations with coefficients in k ; this family depends on the chosen Weierstraß equation but is independent of K . (For a recent discussion concerning families of such equations, see =-=[5]-=-.) If A is an elliptic curve over Q , then one can choose the Weierstraß coefficients αi to be rational integers. The coefficients are essentially unique if we demand that the discriminant of the equa... |

6 |
galoisiennes des points d’ordre fini des courbes elliptiques
- Propriétés
- 1972
(Show Context)
Citation Context ...αL ⊆ L. The group Gn then has an abelian subgroup of index ≤ 2, so it is much smaller than the ambient group GL(2,Z/nZ). In the more common case where A has no complex multiplication, Serre showed in =-=[72]-=- that the index of Gn in GL(2,Z/nZ) is bounded as a function of n. In particular, Gp = GL(2,Z/pZ) for all but finitely many primes p. As background, we point out that the Taniyama-Shimura conjecture w... |

5 |
fields over real quadratic fields and Hecke operators
- Class
- 1972
(Show Context)
Citation Context ... sense when viewed over C .) In [64], I provide a conjectural answer to Serre’s question. Namely, I show that the conjectures made by Serre in [74] imply the following statement, which recalls §10 of =-=[81]-=-. Conjecture 1. Let A be an elliptic curve over C .ThenAismodular if and only if A is isogenous to all elliptic curves σA obtained by conjugating A by algebraic automorphisms σ of the field C . Conjec... |

5 | locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Séminaire Delange-Pisot-Poitou - Facteurs - 1969 |

4 |
On automorphic forms on GL2 and Hecke
- Miyake
- 1971
(Show Context)
Citation Context ...new be the subspace of S(M) spanned by its newforms. Then S(N)= �� � new S(M) � . M|N d| N M We conclude this discussion with the remark that the work of Atkin and Lehner was generalized by T. Miyake =-=[52]-=- in the setting of automorphic forms on GL(2) and then by W. Li [42] in the setting of classical modular forms. 3. Elliptic curves We next introduce some foundational concepts pertaining to elliptic c... |

4 |
Ribet’s Theorem: Shimura-Taniyama-Weil implies Fermat
- Prasad
(Show Context)
Citation Context ...e’s level-lowering conjecture in a 1990 article [62], thereby establishing the implication Conjecture of Taniyama-Shimura =⇒ Fermat’s Last Theorem which was the goal of Frey’s construction. (See also =-=[58]-=- and [63] for expository accounts of this work.) Since the Frey curves associated with Fermat solutions are semistable elliptic curves, I proved that the semistable case of the TaniyamaShimura conject... |

3 |
Available by anonymous ftp from megrez.math.u-bordeaux.fr or math.ucla.edu, in the directory /pub/pari
- Batut, Bernardi, et al.
(Show Context)
Citation Context ...hat unenlightening (see for example [70, §2]), a well-known algorithm of J. Tate [85] makes it possible to compute conductors by hand in specific cases. Alternatively, the computer algebra program gp =-=[3]-=-, the Mathematica package [44] and the Maple package apecs [11] all implement Tate’s algorithm and more generally make it easy to perform elliptic curve calculations on a workstation or personal compu... |

3 |
The Shimura-Taniyama Conjecture (d’apres
- Darmon
(Show Context)
Citation Context ...ticle with Brian Hayes in American Scientist [30], two pieces in the American Mathematical Monthly [13, 28], the report by K. Rubin and A. Silverberg in this Bulletin [68], a long survey by H. Darmon =-=[14]-=-, an article for undergraduates and their teachers by N. Boston [6], and an elementary article by A. Ash and R. Gross [1] which discusses conjectured reciprocity relations between algebraic varieties ... |

3 |
conjectures
- Serre’s
(Show Context)
Citation Context ...e continuous representation ρf with finite image. For details, see [16]. 7. Serre’s conjectures We shall give a brief summary of conjectures made by J.-P. Serre in [74]. A recent article by H. Darmon =-=[15]-=- discusses the conjectures more extensively and emphasizes applications and numerical examples. Let ℓ be a prime number, and let F be an algebraic closure of the prime field Fℓ . Suppose that ρ: Gal(Q... |

3 |
Fermat’s Last Theorem and modern arithmetic, American Scientist 82
- Hayes, Ribet
- 1994
(Show Context)
Citation Context ... expository articles which were written in connection with Wiles’s 1993 announcement. Among these are the author’s news item in the Notices [65] and his article with Brian Hayes in American Scientist =-=[30]-=-, two pieces in the American Mathematical Monthly [13, 28], the report by K. Rubin and A. Silverberg in this Bulletin [68], a long survey by H. Darmon [14], an article for undergraduates and their tea... |