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Galois representations and modular forms
Author(s):
Kenneth
A.
Ribet
Journal:
Bull. Amer. Math. Soc.
32
(1995),
375-402.
MSC (1991):
Primary 11F, 11D
MathSciNet review:
1322785
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References:
- [1]
- A. Ash and R. Gross, Generalized reciprocity laws\,\RM : A context for Wiles{\rm '}s achievement \rm (in preparation).
- [2]
- A?O?L. Atkin and J. Lehner, Hecke operators on $\go m$, Math. Ann. \textbf{185} (1970), 134--160. MR 268123
- [3]
- C. Batut, D. Bernardi, H. Cohen and M. Olivier, GP/PARI, Available by anonymous ftp from {\tt megrez.math.u-bordeaux.fr} or {\tt math.ucla.edu} in the directory {\tt /pub/pari}.
- [4]
- B?J. Birch and W. Kuyk, eds., Modular functions of one variable. \rm IV, Lecture Notes in Math., vol.~476, Springer-Verlag, Berlin and New York, 1975. MR 376533
- [5]
- W. Bosma and H?W. Lenstra, Jr., Complete systems of two addition laws for elliptic curves, J. Number Theory.
- [6]
- N. Boston, A Taylor-made plug for Wiles{\rm '} proof, College Math. J. \textbf{26} (1995), 100--105.
- [7]
- N. Boston and A. Granville, Review of \cite \MvS, Amer. Math. Monthly \textbf{102} (1995), 470--473. MR 1542699
- [8]
- K?M. Buzzard, The levels of modular representations, Cambridge University thesis, 1995.
- [9]
- H. Carayol, Sur les repr\'esentations $\ell $-adiques associ\'ees aux formes modulaires de Hilbert, Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{19} (1986), 409--468. MR 870690
- [10]
- J?H. Coates and S?T. Yau, eds., Elliptic curves and modular forms, Proceedings of a conference held in Hong Kong, December 18--21, 1993, International Press, Cambridge, MA, and Hong Kong.
- [11]
- I. Connell, Apecs \RM (arithmetic of plane elliptic curves\/\RM )\emrule A program written in Maple, Available by anonymous ftp from {\tt math.mcgill.ca} in the directory {\tt /pub/apecs}.
- [12]
- G. Cornell and J. Silverman, eds., Arithmetic geometry, Springer-Verlag, Berlin and New York, 1986. MR 861969
- [13]
- D. Cox, Introduction to Fermat{\rm '}s Last Theorem, Amer. Math. Monthly \textbf{101} (1994), 3--14. MR 1252700
- [14]
- H. Darmon, The Shimura-Taniyama conjecture \RM (d\,\RM 'apr\`es Wiles\/\RM ), Russian Math. Surveys.
- [15]
- H. Darmon, Serre{\rm '}s conjectures, in~\cite \MURTYASED.
- [16]
- P. Deligne and J.-P. Serre, Formes modulaires de poids~$1$, Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{7} (1974), 507--530. MR 379379
- [17]
- F. Diamond, The refined conjecture of Serre, in~\cite \COATESIP. MR 1363493
- [18]
- F. Diamond, On deformation rings and Hecke rings {\rm (submitted)}.
- [19]
- G. Faltings, Endlichkeitss\"atze f\"ur abelsche Variet\"aten \"uber Zahlk\"orpern, Invent. Math. \textbf{73} (1983), 349--366. MR 718935
- [20]
- M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. \textbf{109} (1992), 307--327. MR 1172693
- [21]
- J.-M. Fontaine, Il n{\rm '}y a pas de vari\'et\'e ab\'elienne sur $\Z $, Invent. Math. \textbf{81} (1985), 515--538. MR 807070
- [22]
- J.-M. Fontaine and B. Mazur, Geometric Galois representations, in~\cite \COATESIP. MR 1363495
- [23]
- G. Frey, Links between stable elliptic curves and certain diophantine equations, Ann. Univ. Sarav. Ser. Math. \textbf{1} (1986), 1--40. MR 853387
- [24]
- G. Frey, Links between elliptic curves and solutions of $A-B=C$, J. Indian Math. Soc. \textbf{51} (1987), 117--145. MR 988312
- [25]
- S. Gelbart, Automorphic forms and Artin{\rm '}s conjecture, Lecture Notes in Math. \textbf{627} (1977), 241--276. MR 568306
- [26]
- P. G\'erardin and J?P. Labesse, \emph{The solution to a base change problem for $\GLTWO $ \RM (following Langlands, Saito, Shintani\/\RM )}, Amer. Math. Soc., Providence, RI, 1979 pp.~115--133. MR 546613
- [27]
- F?Q. Gouv\^ea, Deforming Galois representations\,\RM : Controlling the conductor, J. Number Theory \textbf{34} (1990), 95--113. MR 1039770
- [28]
- F?Q. Gouv\^ea, {\rm ``}A marvelous proof\/{\,\rm ''}, Amer. Math. Monthly \textbf{101} (1994), 203--222. MR 1264001
- [29]
- F?Q. Gouv\^ea, Deforming Galois representations\,\RM : A survey, in~\cite \MURTYASED.
- [30]
- B. Hayes and K?A. Ribet, Fermat{\rm '}s Last Theorem and modern arithmetic, Amer. Sci. \textbf{82} (1994), 144--156.
- [31]
- W. R. Hearst III and K?A. Ribet, {\rm Review of ``}Rational points on elliptic curves{\rm ''} \rm by~Joseph~H. Silverman and John T. Tate, Bull. Amer. Math. Soc. (N.S.) \textbf{30} (1994), 248--252.
- [32]
- M. Hindry, {\rm ``}a, b, c{\rm ''}, conducteur, discriminant, Publ. Math. Univ. Pierre et Marie Curie, Probl\`emes diophantiens (1986--87).
- [33]
- A. Jackson, Update on proof of Fermat{\rm '}s Last Theorem, Notices Amer. Math. Soc. \textbf{41} (1994), 185--186. MR 1261593
- [34]
- A. Jackson, Another step toward Fermat, Notices Amer. Math. Soc. \textbf{42} (1995), 48.
- [35]
- A?W.~Knapp, Elliptic curves, Math. Notes, vol. 40, Princeton Univ. Press, Princeton, NJ, 1992. MR 1193029
- [36]
- S. Lang, Introduction to modular forms, Springer-Verlag, Berlin and New York, 1976. MR 429740
- [37]
- S. Lang, Abelian varieties, Springer-Verlag, Berlin and New York, 1983. MR 713430
- [38]
- S. Lang, The \Weil file, Available directly from S.~Lang, Math. Dept., Yale Univ.
- [39]
- S. Lang, Old and new conjectured diophantine inequalities, Bull. Amer. Math. Soc. (N.S.) \textbf{23} (1990), 37--75. MR 1005184
- [40]
- R?P. Langlands, Base change for $\GLTWO $, Ann. Math. Stud., vol. 96, Princeton Univ. Press, Princeton, NJ, 1980. MR 574808
- [41]
- H?W. Lenstra, Jr., Complete intersections and Gorenstein rings, in~\cite \COATESIP.
- [42]
- W.-C? W. Li, Newforms and functional equations, Math. Ann. \textbf{212} (1975), 285--315. MR 369263
- [43]
- H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1986. MR 879273
- [44]
- P?A. van Mulbregt and J?H. Silverman, Elliptic curve calculator, Available by anonymous ftp from {\tt gauss.math.brown.edu} in the directory {\tt /dist/EllipticCurve}.
- [45]
- B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{47} (1977), 33--186. MR 488287
- [46]
- B. Mazur, Rational isogenies of prime degree, Invent. Math. \textbf{44} (1978), 129--162. MR 482230
- [47]
- B. Mazur, \emph{Deforming Galois representations}, Math. Sci. Res. Inst. Publ., vol. 16, Springer-Verlag, Berlin and New York, 1989 pp.~385--437. MR 1012172
- [48]
- B. Mazur, Number theory as gadfly, Amer. Math. Monthly \textbf{98} (1991), 593--610. MR 1121312
- [49]
- B. Mazur, Very rough course notes for Math $257y$, (parts \rm I--III to appear as {\it Galois Deformations and Hecke Curves\/}\rm ).
- [50]
- B. Mazur, \emph{Questions about number}, Cambridge Univ. Press, Cambridge.
- [51]
- B. Mazur and J. Tilouine, Repr\'esentations galoisiennes, diff\'erentielles de K\"ahler et $\ll $conjectures principales$\gg $, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{71} (1990), 9--103. MR 1079644
- [52]
- T. Miyake, On automorphic forms on $\varGLTWO $ and Hecke operators, Ann. Math. \textbf{94} (1971), 174--189. MR 299559
- [53]
- T. Miyake, Modular forms, Springer-Verlag, Berlin and New York, 1989. MR 1021004
- [54]
- V?K. Murty, Introduction to Abelian varieties, CRM Monograph Series, vol.~3, Amer. Math. Soc., Providence, RI, 1993. MR 1231797
- [\MURTYASED , ed.]
- V?K. Murty, Elliptic curves, Galois representations and modular forms, CMS Conf. Proc., Amer. Math. Soc., Providence, RI.
- [56]
- J. Oesterl\'e, Nouvelles approches du {\rm ``}th\'eor\`eme\/{\rm ''} de Fermat, Ast\'erisque \textbf{161/162} (1988), 165--186. MR 992208
- [57]
- A. Ogg, Elliptic curves and wild ramification, Amer. J. Math. \textbf{89} (1967), 1--21. MR 207694
- [58]
- D. Prasad, Ribet{\rm '}s Theorem\/\RM : Shimura-Taniyama-Weil implies Fermat, in~\cite \MURTYASED.
- [59]
- R. Ramakrishna, On a variation of Mazur{\rm '}s deformation functor, Compositio Math. \textbf{87} (1993), 269--286. MR 1227448
- [60]
- K?A. Ribet, The $\ell $-adic representations attached to an eigenform with Nebentypus\,\RM : A survey, Lecture Notes in Math. \textbf{601} (1977), 17--52. MR 453647
- [61]
- K?A. Ribet, \emph{Congruence relations between modular forms} pp.~503--514. MR 804706
- [62]
- K?A. Ribet, On modular representations of $\GalQ $ arising from modular forms, Invent. Math. \textbf{100} (1990), 431--476. MR 1047143
- [63]
- K?A. Ribet, From the \Weil conjecture to Fermat{\rm '}s Last Theorem, Ann. Fac. Sci. Toulouse Math. \textbf{11} (1990), 116--139. MR 1191476
- [64]
- K?A. Ribet, \emph{Abelian varieties over $\Q $ and modular forms}, Korea Adv. Inst. Sci. Tech., Taejon, 1992 pp.~53--79. MR 1212980
- [65]
- K?A. Ribet, Wiles proves Taniyama{\rm '}s conjecture\,\RM ; Fermat{\rm '}s Last Theorem follows, Notices Amer. Math. Soc. \textbf{40} (1993), 575--576. MR 1228162
- [66]
- K?A. Ribet, Modular elliptic curves and Fermat{\rm '}s Last Theorem\/\RM : A lecture presented at George Washington University, Washington, DC, August \rm 1993, Selected Lectures in Math., Amer. Math. Soc., Providence, RI, 1993 (videotape). MR 1243883
- [67]
- K?A. Ribet, \emph{Report on mod~$\ell $ representations of $\GalQ $}, Amer. Math. Soc., Providence, RI, 1994 pp.~639--676. MR 1265566
- [68]
- K. Rubin and A. Silverberg, A report on Wiles{\rm '} Cambridge Lectures, Bull. Amer. Math. Soc. (N.S.) \textbf{31} (1994), 15--38. MR 1256978
- [69]
- K. Rubin and A. Silverberg, Families of elliptic curves with constant mod~$p$ representations, in~\cite \COATESIP. MR 1363500
- [70]
- J.-P. Serre, Facteurs locaux des fonctions z\^eta des vari\'et\'es alg\'ebriques \RM (d\'efinitions et conjectures\RM ), S\'em. Delange-Pisot-Poitou, no. 19 (1969--70), Institut Henri Poincar\'e, Paris, 1970--71.
- [71]
- J.-P. Serre, Propri\'et\'es galoisiennes des points d\,{\rm '}ordre fini des courbes elliptiques, Invent. Math. \textbf{15} (1972), 259--331. MR 387283
- [72]
- J.-P. Serre, A course in arithmetic, Graduate Texts in Math., vol. 7, Springer-Verlag, New York, Heidelberg, and Berlin, 1973. MR 344216
- [73]
- J.-P. Serre, \emph{Lettre \`a J.-F. Mestre {\rm (13} ao\^ut {\rm 1985)}}, Amer. Math. Soc., Providence, RI, 1987 pp.~263--268. MR 902597
- [74]
- J.-P. Serre, Sur les repr\'esentations modulaires de degr\'e~$2$ de $\GalQ $, Duke Math. J. \textbf{54} (1987), 179--230. MR 885783
- [75]
- J.-P. Serre, Algebraic groups and class fields, Graduate Texts in Math., vol. 117, Springer-Verlag, New York, Heidelberg, and Berlin, 1988. MR 918564
- [76]
- J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. \textbf{88} (1968), 492--517. MR 236190
- [77]
- G. Shimura, An $\ell $-adic method in the theory of automorphic forms, {\rm 1968} {\rm (unpublished)}.
- [78]
- G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. \textbf{221} (1966), 209--220. MR 188198
- [79]
- G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, NJ, 1971. MR 314766
- [80]
- G. Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. \textbf{43} (1971), 199--208. MR 296050
- [81]
- G. Shimura, Class fields over real quadratic fields and Hecke operators, Ann. of Math. \textbf{95} (1972), 131--190. MR 314801
- [82]
- G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan \textbf{25} (1973), 523--544. MR 318162
- [83]
- J?H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math., vol.~106, Springer-Verlag, Berlin and New York, 1986. MR 817210
- [84]
- J?H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Math., vol.~151, Springer-Verlag, Berlin and New York, 1994. MR 1312368
- [85]
- J?T. Tate, \emph{Algorithm for determining the type of a singular fiber in an elliptic pencil}, Springer, New York, 1975 pp.~33--52. MR 393039
- [86]
- J?T. Tate, \emph{The non-existence of certain Galois extensions of~$\Bbb Q$ unramified outside~\rm 2} vol.~174, Amer. Math. Soc., Providence, RI, 1994 pp.~153--156. MR 1299740
- [87]
- R?L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. \textbf{141} (1995), 553--572. MR 1333036
- [88]
- J. Tunnell, Artin{\rm '}s conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) \textbf{5} (1981), 173--175. MR 621884
- [89]
- M. vos Savant, The world\,{\rm '}s most famous math problem\/\RM : The proof of Fermat{\rm '}s Last Theorem and other mathematical mysteries, St.~Martin's Press, New York, 1993. MR 1282729
- [90]
- A. Wiles, Modular elliptic curves and Fermat{\rm '}s Last Theorem, Ann. of Math. \textbf{141} (1995), 443--551. MR 1333035
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Additional Information:
DOI:
10.1090/S0273-0979-1995-00616-6
PII:
S 0273-0979(1995)00616-6
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