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Galois representations. (English) Zbl 1074.11030
The author states and explains very precisely the most important conjectures in the areas of Galois representations and arithmetic respectively automorphic \(L\)-functions. This explanation (an expansion of the author’s ICM-talk from Beijing) is mainly meant to serve the general mathematical audience as a guideline. Some recent results are outlined in the final chapter.

MSC:
11F80 Galois representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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[1] Artin ( E. ) , Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren , Abh. Math. Sem. Univ. Hamburg 8 , p. 292 - 306 ( 1930 ). JFM 56.0173.02 · JFM 56.0173.02
[2] Arthur ( J. ) and Clozel ( L. ) , Simple algebras, base change and the advanced theory of the trace formula , Annals of Math. Studies 120 , PUP ( 1989 ). MR 1007299 | Zbl 0682.10022 · Zbl 0682.10022
[3] Berger ( L. ) , Représentations p-adiques et équations différentielles , Invent. math. 148 , p. 219 - 284 ( 2002 ). MR 1906150 | Zbl 01777245 · Zbl 1113.14016 · doi:10.1007/s002220100202
[4] Berthelot ( P. ) , Altérations de variétés algébriques (d’après A.J.de Jong , Astérisque 241 , p. 273 - 311 ( 1997 ). Numdam | MR 1472543 | Zbl 0924.14007 · Zbl 0924.14007 · numdam:SB_1995-1996__38__273_0 · eudml:110217
[5] Breuil ( C. ) , Conrad ( B. ) , Diamond ( F. ) and Taylor ( R. ) , On the modularity of elliptic curves over Q , J.A.M.S . 14 , p. 843 - 939 ( 2001 ). MR 1839918 | Zbl 0982.11033 · Zbl 0982.11033 · doi:10.1090/S0894-0347-01-00370-8
[6] Bump ( D. ) , Friedberg ( S. ) and Hoffstein ( J. ) , Nonvanishing theorems for L-functions of modular forms and their derivatives , Invent. math. 102 , p. 543 - 618 ( 1990 ). MR 1074487 | Zbl 0721.11023 · Zbl 0721.11023 · doi:10.1007/BF01233440 · eudml:143846
[7] Bloch ( S. ) and Kato ( K. ) , L-functions and Tamagawa numbers of motives , in ” The Grothendieck Festschrift I ”, Birkauser ( 1990 ). MR 1086888 | Zbl 0768.14001 · Zbl 0768.14001
[8] Brauer ( R. ) , On Artin’s L-series with general group characters , Ann. Math. ( 2 ) 48 , p. 502 - 514 ( 1947 ). MR 20105 | Zbl 0029.01503 · Zbl 0029.01503 · doi:10.2307/1969183
[9] Birch ( B. ) and Swinnerton-Dyer ( P. ) , Notes on elliptic curves II , J. Reine Angew. Math. 218 , p. 79 - 108 ( 1965 ). Article | MR 179168 | Zbl 0147.02506 · Zbl 0147.02506 · doi:10.1515/crll.1965.218.79 · crelle:GDZPPN002181169 · eudml:150676
[10] Buzzard ( K. ) and Taylor ( R. ) , Companion forms and weight one forms , Annals of math 149 , p. 905 - 919 ( 1999 ). MR 1709306 | Zbl 0965.11019 · Zbl 0965.11019 · doi:10.2307/121076 · www.math.princeton.edu · eudml:121093
[11] Conrad ( B. ) , Diamond ( F. ) and Taylor ( R. ) , Modularity of certain potentially Barsotti-Tate Galois representations , JAMS 12 , p. 521 - 567 ( 1999 ). MR 1639612 | Zbl 0923.11085 · Zbl 0923.11085 · doi:10.1090/S0894-0347-99-00287-8
[12] Clozel ( L. ) , Motifs et formes automorphes : applications du principe de fonctorialité, in ”Automorphic forms, Shimura varieties and L-functions I” , Academic Press ( 1990 ). MR 1044819 | Zbl 0705.11029 · Zbl 0705.11029
[13] Clozel ( L. ) , Représentations Galoisiennes associées aux representations automorphes autoduales de GL(n , Pub. Math. IHES 73 , p. 97 - 145 ( 1991 ). Numdam | MR 1114211 | Zbl 0739.11020 · Zbl 0739.11020 · doi:10.1007/BF02699257 · numdam:PMIHES_1991__73__97_0 · eudml:104076
[14] Cogdell ( J. ) and Piatetski-Shapiro ( I. ) , Converse theorems for GLn , Publ. Math. IHES 79 , p. 157 - 214 ( 1994 ). Numdam | MR 1307299 | Zbl 0814.11033 · Zbl 0814.11033 · doi:10.1007/BF02698889 · numdam:PMIHES_1994__79__157_0 · eudml:104095
[15] Cogdell ( J. ) and Piatetski-Shapiro ( I. ) , Converse theorems for GLn II , J. reine angew. Math. 507 , p. 165 - 188 ( 1999 ). MR 1670207 | Zbl 0912.11022 · Zbl 0912.11022 · doi:10.1515/crll.1999.507.165
[16] Deligne ( P. ) , Les constantes des équations fonctionnelles des fonctions L, in LNM 349 , Springer ( 1973 ). MR 349635 | Zbl 0271.14011 · Zbl 0271.14011
[17] Deligne ( P. ) , Courbes elliptiques : formulaire d’après Tate, in ”Modular functions of one variable IV”, LNM 476 , Springer ( 1975 ). MR 387292 · Zbl 1214.11075 · doi:10.1007/BFb0097583
[18] Deligne ( P. ) , La conjecture de Weil I , Publ. Math. IHES 43 , p. 273 - 307 ( 1974 ). Numdam | MR 340258 · Zbl 0287.14001 · numdam:PMIHES_1974__43__273_0
[19] Diamond ( F. ) , The Taylor-Wiles construction and multiplicity one , Invent. Math. 128 , p. 379 - 391 ( 1997 ). MR 1440309 | Zbl 0916.11037 · Zbl 0916.11037 · doi:10.1007/s002220050144
[20] Dixniier ( J. ) , Algèbres enveloppantes , Gauthier Villars ( 1974 ). · Zbl 0308.17007
[21] Drinfeld ( V. ) , Proof of the global Langlands conjecture for GL(2) over a function field , Funct. anal. and its appl. 11 , p. 223 - 225 ( 1977 ). MR 498489 | Zbl 0423.12010 · Zbl 0423.12010 · doi:10.1007/BF01079470
[22] Faltings ( G. ) , Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , Invent. math. 73 , p. 349 - 366 ( 1983 ). Zbl 0588.14026 · Zbl 0588.14026 · doi:10.1007/BF01388432 · eudml:143051
[23] Fontaine ( J.-M. ) , talk at ” Mathematische Arbeitstagung 1988 ”, Maz-Planck-Institut für Mathematik preprint no. 30 of 1988 .
[24] Fontaine ( J.-M. ) , Le corps des périodes p-adiques , Astérisque 223 , p. 59 - 111 ( 1994 ). MR 1293971 | Zbl 0940.14012 · Zbl 0940.14012
[25] Fontaine ( J.-M. ) , Représentations p-adiques semi-stables , Astérisque 223 , p. 113 - 184 ( 1994 ). MR 1293972 | Zbl 0865.14009 · Zbl 0865.14009
[26] Fontaine ( J.-M. ) , Représentations l-adiques potentiellement semi-stables , Astérisque 223 , p. 321 - 347 ( 1994 ). MR 1293977 | Zbl 0873.14020 · Zbl 0873.14020
[27] Fontaine ( J.-M. ) and Mazur ( B. ) , Geometric Galois representations, in Elliptic curves, modular forms and Fermat’s last theorem (J.Coates and S.-T.Yau eds.), International Press ( 1995 ). MR 1363495 | Zbl 0839.14011 · Zbl 0839.14011
[28] Grothendieck ( A. ) , Formule de Lefschetz et rationalité des fonctions L , in ” Dix exposés sur la cohomologie des schémas ”, North-Holland ( 1968 ). MR 476737 | Zbl 0197.47102 · Zbl 0197.47102
[29] Gelbart ( S. ) and Jacquet ( H. ) , A relation between automorphic representations of GL(2) and GL(3 , Ann. Sci. ENS 11 , p. 471 - 552 ( 1978 ). Numdam | MR 533066 | Zbl 0406.10022 · Zbl 0406.10022 · numdam:ASENS_1978_4_11_4_471_0 · eudml:82024
[30] Godenient and Jacquet ( H. ) , Zeta functions of simple algebras, LNM 260 , Springer ( 1972 ). MR 374100 | Zbl 0244.12011 · Zbl 0244.12011 · doi:10.1007/BFb0070263
[31] Gross ( B. ) and Zagier ( D. ) , Heegner points and derivatives of L-series , Invent. math. 84 , p. 225 - 320 ( 1986 ). MR 833192 | Zbl 0608.14019 · Zbl 0608.14019 · doi:10.1007/BF01388809 · eudml:143341
[32] Harris ( M. ) , The local Langlands conjecture for GL(n) over a p-adic field, n < p , Invent. Math. 134 , p. 177 - 210 ( 1998 ). Zbl 0921.11060 · Zbl 0921.11060 · doi:10.1007/s002220050262
[33] Hecke ( E. ) , Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, II , Math. Ann. 114 , p. 1 - 28 and p. 316 - 351 ( 1937 ). MR 1513122 | Zbl 0015.40202 | JFM 63.0339.03 · Zbl 0015.40202 · doi:10.1007/BF01594160 · eudml:159899
[34] Henniart ( G. ) , On the local Langlands conjecture for GL(n): the cyclic case , Ann. of Math. 123 , 145 - 203 ( 1986 ). MR 825841 | Zbl 0588.12010 · Zbl 0588.12010 · doi:10.2307/1971354
[35] Henniart ( G. ) , Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adiques , Invent. math. 139 , p. 439 - 455 ( 2000 ). MR 1738446 | Zbl 1048.11092 · Zbl 1048.11092 · doi:10.1007/s002220050012
[36] Herbrand ( J. ) , Sur les classes des corps circulaires , J. Math. Pures Appl. 11 , p. 417 - 441 ( 1932 ). Article | JFM 58.0180.02 · JFM 58.0180.02 · minidml.mathdoc.fr
[37] Harris ( M. ) and Taylor ( R. ) , The geometry and cohomology of some simple Shimura varieties , PUP ( 2001 ). Zbl 1036.11027 · Zbl 1036.11027
[38] Ihara ( Y. ) , On modular curves over finite fields , in ” Proceedings of an international colloquium on discrete subgroups of Lie groups and applications to moduli ” OUP ( 1975 ). MR 399105 | Zbl 0343.14007 · Zbl 0343.14007
[39] Illusie ( L. ) , Cohomologie de de Rham et cohomologie étale p-adique , Astérisque 189 - 190 , p. 325 - 374 ( 1990 ). Numdam | MR 1099881 · Zbl 0736.14005 · numdam:SB_1989-1990__32__325_0 · eudml:110129
[40] Jacquet ( H. ) , Principal L-functions of the linear group , in ” Automorphic forms, representations and L-functions 2 ” AMS ( 1979 ). MR 546609 | Zbl 0413.12007 · Zbl 0413.12007
[41] Jacquet ( H. ) and Langlands ( R. ) , Automorphic forms on GL(2), LNM 114 , Springer 1970 . MR 401654 | Zbl 0236.12010 · Zbl 0236.12010 · doi:10.1007/BFb0058988
[42] Jacquet ( H. ) , Piatetski-Shapiro ( I. ) and Shalika ( J. ) , Automorphic forms on GL(3 , Annals of math . 109 , p. 169 - 258 ( 1979 ). MR 519356 | Zbl 0401.10037 · Zbl 0401.10037 · doi:10.2307/1971270
[43] Jacquet ( H. ) , Piatetski-Shapiro ( I. ) and Shalika ( J. ) , Relèevement cubique non normal , C. R. Acad. Sci. Paris Sér. I Math. 292 no. 12 , p. 567 - 571 ( 1981 ). MR 615450 | Zbl 0475.12017 · Zbl 0475.12017
[44] Jacquet ( H. ) and Shalika ( J. ) , On Euler products and the classification of automorphic representations II , Amer. J. Math. 103 , p. 777 - 815 ( 1981 ). MR 623137 | Zbl 0491.10020 · Zbl 0491.10020 · doi:10.2307/2374050
[45] Kolyvagin ( V. ) , Finiteness of E(Q) and Sh(E, Q) for a subclass of Weil curves , Izv. Akad. Nauk SSSR Ser. Mat. 52 , p. 522 - 540 ( 1988 ). MR 954295 | Zbl 0662.14017 · Zbl 0662.14017 · doi:10.1070/IM1989v032n03ABEH000779
[46] Kolyvagin ( V. ) , On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves , in ”Proceedings of the Kyoto International Congress of Mathematicians” Math. Soc. Japan ( 1991 ). MR 1159231 | Zbl 0749.14012 · Zbl 0749.14012
[47] Kottwitz ( R. ) , On the \lambda -adic representations associated to some simple Shimura varieties , Invent. math. 108 , p. 653 - 665 ( 1992 ). MR 1163241 | Zbl 0765.22011 · Zbl 0765.22011 · doi:10.1007/BF02100620 · eudml:144008
[48] Lafforgue ( L. ) , Chtoukas de Drinfeld et correspondance de Langlands , Invent. math. 147 , p. 1 - 241 ( 2002 ). MR 1875184 | Zbl 1038.11075 · Zbl 1038.11075 · doi:10.1007/s002220100174
[49] Langlands ( R.P. ) , On the classification of representations of real reductive groups , mimeographed notes from the IAS, ( 1973 ).
[50] Langlands ( R.P. ) , Automorphic representations, Shimura varieties and motives. Ein Märchen . in ” Automorphic forms, representations and L-functions II ” AMS ( 1979 ). Zbl 0447.12009 · Zbl 0447.12009
[51] Langlands ( R.P. ) , Base change for GL(2 , Annals of Math. Studies 96 , Princeton Univ. Press , Princeton , ( 1980 ). MR 574808 | Zbl 0444.22007 · Zbl 0444.22007
[52] Laumon ( G. ) , Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil , Publ. math. IHES 65 , p. 131 - 210 ( 1987 ). Numdam | MR 908218 | Zbl 0641.14009 · Zbl 0641.14009 · doi:10.1007/BF02698937 · numdam:PMIHES_1987__65__131_0 · eudml:104019
[53] Mann ( W.R. ) , Local level raising for GLn , preprint.
[54] Moret-Bailly ( L. ) , Groupes de Picard et problèmes de Skolem II , Ann. Sci. ENS 22 , p. 181 - 194 ( 1989 ). Numdam | MR 1005158 | Zbl 0704.14015 · Zbl 0704.14015 · numdam:ASENS_1989_4_22_2_181_0 · eudml:82251
[55] Miyake ( T. ) , Modular forms , Springer ( 1989 ). MR 1021004 | Zbl 0701.11014 · Zbl 0701.11014
[56] Murty ( R. ) and Murty ( K. ) , Mean values of derivatives of modular L-series , Ann. of Math. ( 2 ) 133 , p. 447 - 475 ( 1991 ). MR 1109350 | Zbl 0745.11032 · Zbl 0745.11032 · doi:10.2307/2944316
[57] Pop ( F. ) , Embedding problems over large fields , Annals of Math. 144 , p. 1 - 34 ( 1996 ). MR 1405941 | Zbl 0862.12003 · Zbl 0862.12003 · doi:10.2307/2118581
[58] Piatetski-Shapiro ( I. ) , Zeta functions of GL(n , Univ. Maryland Techn. Rep. TR 76 - 46 ( 1976 ).
[59] Ribet ( K. ) , A modular construction of unramified p-extensions of Q(\mu p ), Invent. math. 34 , p. 151 - 162 ( 1976 ). MR 419403 | Zbl 0338.12003 · Zbl 0338.12003 · doi:10.1007/BF01403065 · eudml:142394
[60] Ribet ( K. ) , Congruence relations between modular forms , in ”Proceedings of the Warsaw ICM” Polish Scientific Publishers ( 1984 ). MR 804706 | Zbl 0575.10024 · Zbl 0575.10024
[61] Savitt ( D. ) , Modularity of some potentially Barsotti-Tate Galois representations , preprint. arXiv | MR 2004122 · Zbl 1053.11048 · minidml.mathdoc.fr
[62] Serre ( J.-P. ) , Corps locaux , Hermann ( 1962 ). MR 354618 · Zbl 0137.02601
[63] Serre ( J.-P. ) , Abelian l-adic representations and elliptic curves , Benjamin ( 1968 ). MR 263823 | Zbl 0186.25701 · Zbl 0186.25701
[64] Artin ( M. ) , Grothendieck ( A. ) and Verdier ( J.L. ) , Théorie des topos et cohomologie étale des schémas, LNM 269 and 270 and 305 , Springer ( 1972 / 73 ). MR 354653
[65] Illusie ( L. ) , Cohomologie l-adique et fonctions L, LNM 589 Springer ( 1977 ). MR 491704 · Zbl 0352.14009
[66] Skinner ( C. ) and Wiles ( A. ) , Residually reducible representations and modular forms , Inst. Hautes Etudes Sci. Publ. Math. 89 , p. 5 - 126 ( 1999 ). Numdam | MR 1793414 | Zbl 1005.11030 · Zbl 1005.11030 · doi:10.1007/BF02698855 · numdam:PMIHES_1999__89__5_0 · eudml:104160
[67] Skinner ( C. ) and Wiles ( A. ) , Base change and a problem of Serre , Duke Math. J. 107 , p. 15 - 25 ( 2001 ). Article | MR 1815248 | Zbl 1016.11017 · Zbl 1016.11017 · doi:10.1215/S0012-7094-01-10712-6 · minidml.mathdoc.fr
[68] Skinner ( C. ) and Wiles ( A. ) , Nearly ordinary deformations of irreducible residual representations , Annales de la Faculté des Sciences de Toulouse X , p. 185 - 215 ( 2001 ). Numdam | MR 1928993 | Zbl 1024.11036 · Zbl 1024.11036 · doi:10.5802/afst.988 · numdam:AFST_2001_6_10_1_185_0 · eudml:73538
[69] Tate ( J. ) , Number theoretic background , in A. Borel and W.Casselman ” Automorphic forms, representations and L-functions ” , Proc. Symposia in Pure Math. 33 ( 2 ), AMS ( 1979 ). MR 546607 | Zbl 0422.12007 · Zbl 0422.12007
[70] Taylor ( R. ) , On the meromorphic continuation of degree two L-functions , preprint available at http://www.math.harvard.edu/ rtaylor . · Zbl 1138.11051
[71] Tunnell ( J. ) , Artin’s conjecture for representations of octahedral type , Bull. AMS 5 , p. 173 - 175 ( 1981 ). Article | MR 621884 | Zbl 0475.12016 · Zbl 0475.12016 · doi:10.1090/S0273-0979-1981-14936-3 · minidml.mathdoc.fr
[72] Taylor ( R. ) and Wiles ( A. ) , Ring theoretic properties of certain Hecke algebras , Annals of Math. 141 , p. 553 - 572 ( 1995 ). MR 1333036 | Zbl 0823.11030 · Zbl 0823.11030 · doi:10.2307/2118560
[73] Van Der Waall ( R. ) , On Brauer’s induction formula of characters of groups , Arch. Math. 63 , p. 208 - 210 ( 1994 ). MR 1287248 | Zbl 0842.20010 · Zbl 0842.20010 · doi:10.1007/BF01189821
[74] Weil ( A. ) , Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , Math. Ann. 168 , p. 149 - 156 ( 1967 ). MR 207658 | Zbl 0158.08601 · Zbl 0158.08601 · doi:10.1007/BF01361551 · eudml:161499
[75] Wiles ( A. ) , Modular elliptic curves and Fermat’s last theorem , Annals of Math. 141 , 443 - 551 ( 1995 ). MR 1333035 | Zbl 0823.11029 · Zbl 0823.11029 · doi:10.2307/2118559
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