×

zbMATH — the first resource for mathematics

La conjecture de Langlands locale pour GL(3). (The local Langlands conjecture for GL(3)). (French) Zbl 0577.12011
In this paper the local Langlands conjecture for GL(3) is proved: Let F be a non-Archimedean local field. There is a unique map from the set of equivalence classes of \(\Phi\)-semisimple complex representations of degree 3 of the Weil-Deligne group of F to the set of equivalence classes of irreducible admissible representations of GL(3,F) preserving L- and \(\epsilon\)-factors. The proof of this theorem and of the properties of the map can be reduced to the proof of the existence, uniqueness and bijectivity of a map from the subset of irreducible representations of the Weil group to the subset of supercuspidal representations of GL(3,F), preserving \(\epsilon\)-factors and commuting with torsion by characters of \(F^{\times}.\)
By known global results the existence of the supercuspidal representation corresponding to a given irreducible representation of the Weil group is ensured in all cases except when F is an extension of \({\mathbb{Q}}_ 3\) and the representation is primitive. This difficult case is treated by the author using a tame base change. It is necessary to have an explicit description of all irreducible supercuspidal representations of GL(3,F); for this, the technique of Carayol is used, and in order to prove that the description is complete, the author uses a bijection between the irreducible supercuspidal representation of GL(3,F) and the irreducible admissible representations of the multiplicative group of a central division algebra of dimension 9 over F. A proof of the local Langlands conjecture for GL(2) is also indicated, and used in the proof for GL(3).
Reviewer: J.G.M.Mars

MSC:
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] J. ARTHUR : Eisenstein series and the trace formula , in [Co], Part , pp. 253-274. MR 81b:10020 | Zbl 0431.22016 · Zbl 0431.22016
[2] J. Arthur : A trace formula for reductive groups I : terms associated to classes in G(q) , Duke Math. J. vol. 45 no 4, pp. 911-952 ( 1978 ). Article | MR 80d:10043 | Zbl 0499.10032 · Zbl 0499.10032 · doi:10.1215/S0012-7094-78-04542-8 · minidml.mathdoc.fr
[3] A. Borel et alii : Seminar on algebraic groups and related finite groups , Lecture Notes in Mathematics no 131, Srpinger-Verlag ( 1972 ). Zbl 0192.36201 · Zbl 0192.36201 · doi:10.1007/BFb0081541
[4] H. Carayol : Representations cuspidales du groupe linèc)aire , prèc)publication Universitèc) Paris VII ( 1982 ).
[5] P. Cartier : Reprèc)sentations of P-adic groups : a survey , in [Co], Part 1, pp. 111-156. Zbl 0421.22010 · Zbl 0421.22010
[6] P. Cartier : La conjecture locale de Langlands pour Gl(2) et la dèc)monstration de Ph. Kutzko , Sèc)minaire Bourbaki 1979 / 1980 , exposèc) no 550, Lect. Notes in Math. no 842, Springer-Verlag ( 1981 ). Numdam | Zbl 0498.12013 · Zbl 0498.12013 · numdam:SB_1979-1980__22__112_0 · eudml:109948
[7] P. Cartier : Fonction L d’Artin : thèc)orie locale (Cours 1977 / 1978 rèc)digèc) par Guy Henniart), prèc)publicaiton IHES ( 1980 ).
[8] W. Casselman : Introduction to the theory admissible representation of P-adic reductive groups , notes polycopièc)es ( 1977 ).
[9] W. Casselman : Automorphic forms, representations and L-functions , Proc. Sym. pure Math. vol. 33, AMS, Providence ( 1979 ). · Zbl 0403.00003
[10] P. Deligne : Les constantes des èc)quations fonctionnelles des fonctions L , in Modular functions of one variable II, Lect. Notes in Math. no 349, Springer-Verlag ( 1973 ). Zbl 0271.14011 · Zbl 0271.14011
[11] P. Deligne : Formes modulaires et reprèc)sentations de GL(2) , in Modular functions of one variable II, Lect. Notes in Math. no 349, Springer-Verlag ( 1973 ). Zbl 0271.10032 · Zbl 0271.10032
[12] P. Deligne : Lettre à R. P. Langlands, datèc)e du 21 mars 1977. [DH] P. Deligne et G. Henniart : Sur la variation, par torsion, des constantes locales d’èc)quations fonctionnelles de fonctions L , Inv. Math. 64, pp. 89-118 ( 1981 ). MR 83e:12013 | Zbl 0442.12012 · Zbl 0442.12012 · doi:10.1007/BF01393935 · eudml:142815
[13] P. Deligne et D. Kazhdan : On representation of local division algebras , prèc)publication 1982 .
[14] P. Deligne et G. Lusztig : Representations of reductive groups over finite fields , Ann. of Math. 103, pp. 103-161 ( 1976 ). MR 52 #14076 | Zbl 0336.20029 · Zbl 0336.20029 · doi:10.2307/1971021
[15] D. Flath : A comparison of the automorphic representation of GL(3) and its twisted forms , Pacific J. of Math. 97, pp. 373-402. Article | MR 83d:22013 | Zbl 0488.22032 · Zbl 0488.22032 · doi:10.2140/pjm.1981.97.373 · minidml.mathdoc.fr
[16] Y. Flicker : The trace formula and base change for GL(3) , Lect. Notes in Math. no 927, Springer Verlag ( 1982 ). MR 84d:10035 | Zbl 0481.10023 · Zbl 0481.10023 · doi:10.1007/BFb0094272
[17] S. Gelbart et H. Jacquet : A relation between automorphic representaton of GL(2) and GL(3) , Ann. Scient. Ec. Norm. 4ème sèc)rie, t. 11, pp. 471-542 ( 1978 ). Numdam | MR 81e:10025 | Zbl 0406.10022 · Zbl 0406.10022 · numdam:ASENS_1978_4_11_4_471_0 · eudml:82024
[18] I. Gel’fand M. Graev I. Pijatetskki-Shapiro : Representation theory and automorphic functions , W. B. Saunders Company, Philadelphia ( 1969 ). Zbl 0177.18003 · Zbl 0177.18003
[19] P. Gèc)rardin : Cuspidal unramified series for central simple algebras over local fields , in [Co], Part I, pp. 157-170. [Ge2] P. Gèc)rardin : Weil representations associated to finite fields , Journal of Algebra, vol. 46, no 1, pp. 54-101 ( 1977 ). MR 57 #470 | Zbl 0359.20008 · Zbl 0359.20008 · doi:10.1016/0021-8693(77)90394-5
[20] P. Gèc)rardin et Ph Kutzko : Facteurs locaux pour GL(2) , Ann. Scient. E.N.S., t. 13, pp. 349-384. Numdam | MR 82i:22020 | Zbl 0448.22015 · Zbl 0448.22015 · numdam:ASENS_1980_4_13_3_349_0 · eudml:82055
[21] P. Gèc)rardin et J. P. Labesse : The solution of a base change problem for GL(2) , in [Co], Part II, pp. 115-133. Zbl 0412.10018 · Zbl 0412.10018
[22] R. Godement : Domaines fondamentaux des groupes arithmèc)tiques , sèc)minaire Bourbaki no 257 ( 1963 ). Numdam | Zbl 0136.30101 · Zbl 0136.30101 · numdam:SB_1962-1964__8__201_0 · eudml:109658
[23] R. Godement et H. Jacquet : Zeta functions of simple algebras , Lect. Notes in Math. no 260, Springer-Verlag ( 1972 ). MR 49 #7241 | Zbl 0244.12011 · Zbl 0244.12011 · doi:10.1007/BFb0070263
[24] G. Harder : Minkowskische Reduktionstheorie à\pm /4ber Funktionenkörpern , Inv. Math. 7, pp. 33-54 ( 1969 ). MR 44 #1667 | Zbl 0242.20046 · Zbl 0242.20046 · doi:10.1007/BF01418773 · eudml:141952
[25] Harish-Chandra : Harmonic analysis on reductive p-adic groups , notes by G. van Dijk, Lect. Notes in Math. no 162, Springer-Verlag ( 1970 ). MR 54 #2889 | Zbl 0202.41101 · Zbl 0202.41101
[26] G. Henniart : Reprèc)sentation du groupe de Weil d’un corps local , l’Enseign. Math. t. 26, pp. 155-172 ( 1980 ). MR 81j:12012 | Zbl 0452.12006 · Zbl 0452.12006
[27] R. Howe : The Fourier transform and g erms of characters , (case of GLn over a p-adic field), Math. Ann. 208, pp. 305-322 ( 1974 ). MR 49 #7391 | Zbl 0266.43007 · Zbl 0266.43007 · doi:10.1007/BF01432155 · eudml:162577
[28] R. Howe : Addendum à l’article prèc)cèc)dent , à paraèr)tre.
[29] H. Jacquet : Principal L-functions of the linear group , in [Co], Part 2, pp. 63-86. MR 81f:22029 | Zbl 0413.12007 · Zbl 0413.12007
[30] H. Jacquet : Generic representation , in Non commutative harmonic analysis, Lect. Notes in Math. no 587, Springer-Verlag ( 1977 ). MR 58 #16985 | Zbl 0357.22010 · Zbl 0357.22010
[31] H. Jacquet : Lettre à l’auteur , datée du 13 mars 1981 .
[32] H. Jacquet R.P. Langlands : Automorphic forms on GL(2) , Lect. Notes in Math. n^\circ 114, Springer-Verlag ( 1970 ). MR 53 #5481 | Zbl 0236.12010 · Zbl 0236.12010 · doi:10.1007/BFb0058988
[33] H. Jacquet I. Piztetski-Shapiro J.A. Shalika : Automorphic forms on GL(3) , Ann. of Math. n^\circ 109, pp. 169-258 ( 1979 ). MR 80i:10034a | Zbl 0401.10037 · Zbl 0401.10037 · doi:10.2307/1971270
[34] - id - : Facteurs L et \epsilon du groupe linéaire C.R.A.S. Paris t. 289, pp. 59-61 ( 1979 ).
[35] - id - : Conducteur des représentations du groupe linéaire , Math. Ann. 256, pp. 199-214 ( 1981 ). MR 83c:22025 | Zbl 0443.22013 · Zbl 0443.22013 · doi:10.1007/BF01450798 · eudml:163506
[36] - id - : Facteurs L et \epsilon du groupe linéaire : théorie archimédienne , C.R.A.S. Paris t. 293, pp. 13-18 ( 1981 ).
[37] - id - : Relèvement cubique non normal , C.R.A.S. Paris t. 292, pp. 567-571 ( 1981 ). MR 82i:10035 | Zbl 0475.12017 · Zbl 0475.12017
[38] H. Jacquet : J.A. Shalika : On Euler products and the classification of automorphic I, II , Am. J. of Math. vol. 103, pp. 499-558 et pp. 777-815 ( 1981 ). MR 82m:10050b | Zbl 0491.10020 · Zbl 0491.10020 · doi:10.2307/2374050
[39] H. Koch : Classification of the primitive representations of the Galois group of local fields , Inv. Math. 40, pp. 195-216 ( 1977 ). MR 56 #8540 | Zbl 0376.12003 · Zbl 0376.12003 · doi:10.1007/BF01390345 · eudml:142478
[40] H. Koch : On the local Langlands conjecture for central division algebras of index p , Inv. Math. 62, pp. 243-268 ( 1980 ). MR 82c:12013 | Zbl 0449.12009 · Zbl 0449.12009 · doi:10.1007/BF01389160 · eudml:142772
[41] H. Koch E.W. Zink : Zur Korrespondenz von Darstellungen der Galois-gruppen und der zentralen Divisionalgebren über lokalen Körpern , Math. Nachr. 98, pp. 83-119 ( 1980 ). MR 83h:12025 | Zbl 0475.12024 · Zbl 0475.12024 · doi:10.1002/mana.19800980110
[42] Ph. Kutzko : The irreductible imprimitive local Galois representations of prime dimension , J. of Algebra 57, pp. 101-110 ( 1979 ). Zbl 0438.12007 · Zbl 0438.12007 · doi:10.1016/0021-8693(79)90211-4
[43] Ph. Kutzko : The Langlands conjecture for GL(2) of a local field Ann. of Math. 112, pp. 381-412 ( 1980 ). MR 82e:12019 | Zbl 0469.22013 · Zbl 0469.22013 · doi:10.2307/1971151
[44] S. Lang : Algebraic Number Theory , Addison-Wesley, Reading, 1970 . MR 44 #181 | Zbl 0211.38404 · Zbl 0211.38404
[45] R.P. Langlands : On the classification of irreducible representations of real reductive groups , notes miméographiées, I.A.S., Princeton ( 1973 ).
[46] R.P. Langlands : On the functional equations satisfied by Eisenstein series , Lect. Notes in Math. n^\circ 544, Springer-Verlag ( 1976 ). MR 58 #28319 | Zbl 0332.10018 · Zbl 0332.10018 · doi:10.1007/BFb0079929
[47] R.P. Langlands : Base change for GL(2) , Ann. of Math. studies n^\circ 93, Princeton ( 1980 ). MR 82a:10032 | Zbl 0444.22007 · Zbl 0444.22007
[48] G.A. Miller H.F. Blichtfeld L.E. Dickson : Finite groups of linear homogeneous transformations , (by H.F. Blichtfeld) in Theory and applications of finite groups, Dover, New York ( 1961 ).
[49] A. Moy : Local constants and the tame Langlands correspondence Ph. D. Dissertation, Chicago ( 1982 ). · Zbl 0597.12019
[50] J. Rogawski : An application of the building to orbital integrals , Comp. Math. 42, pp. 417-423 ( 1981 ). Numdam | MR 83g:22011 | Zbl 0471.22020 · Zbl 0471.22020 · numdam:CM_1980__42_3_417_0 · eudml:89488
[51] J. Rogawski : Representations of GL(n) and division algebras over a p-adic field , prépublication ( 1981 ).
[52] J.-P. Serre : Corps locaux , Hermann, Paris ( 1968 ). MR 50 #7096 | Zbl 0137.02601 · Zbl 0137.02601
[53] J.-P. Serre : Représentations linéaires des groupes finis , Hermann, Paris ( 1971 ). MR 50 #4718 | Zbl 0223.20003 · Zbl 0223.20003
[54] J. Shalika : A theorem on semi-simple p-adic groups , Ann. of Math. 95, pp. 226-242 ( 1972 ). MR 48 #2310 | Zbl 0281.22011 · Zbl 0281.22011 · doi:10.2307/1970797
[55] J. Tate : Number theoretic background in [Co] , Part II, pp. 3-26 MR 80m:12009 | Zbl 0422.12007 · Zbl 0422.12007
[56] J.B. Tunnell : On the local Langlands conjecture for GL(2) , Inv. Math. 46, pp. 179-200 ( 1978 ). MR 57 #16262 | Zbl 0385.12006 · Zbl 0385.12006 · doi:10.1007/BF01393255 · eudml:142559
[57] J.B. Tunnell : Report on the local Langlands conjecture for GL(2) , in [Co], Part II, pp. 135-138. MR 80k:12025 | Zbl 0409.12026 · Zbl 0409.12026
[58] J.B. Tunnell : Artin’s conjecture for representations of octahedral type B.A.M.S. vol. 5 , n^\circ 2, pp. 173-175 ( 1981 ). Article | MR 82j:12015 | Zbl 0475.12016 · Zbl 0475.12016 · doi:10.1090/S0273-0979-1981-14936-3 · minidml.mathdoc.fr
[59] M.F. Vigneras : Représentation des algèbres centrales simples sur un corps local non archimédien , prépublication ( 1982 ).
[60] G. Warner : Selberg’s trace formula for non-uniform lattices : the R- rank one case, in Studies in Algebra and number theory, Advances in Math. Suppl. studies, vol. 6, pp. 1-141 ( 1979 ). MR 81f:10044 | Zbl 0466.10018 · Zbl 0466.10018
[61] A. Weil : Basic number theory , Springer-Verlag, Berlin ( 1973 ). Zbl 0267.12001 · Zbl 0267.12001
[62] A. Weil : Sur les courbes algébriques et les variétés qui s’en s’en déduisent . Hermann, Paris, 1948 . MR 10,262c | Zbl 0036.16001 · Zbl 0036.16001
[63] A.V. Zelevinsky : Induced representations for reductive p-adic groups II . On irreducible representations of GL(n), Ann. Scient. E.N.S. 4ème série, t. 13, pp. 165-210 ( 1980 ). Numdam | MR 83g:22012 | Zbl 0441.22014 · Zbl 0441.22014 · numdam:ASENS_1980_4_13_2_165_0 · eudml:82048
[64] E.-W. Zink : Counting primitive projective representation of local Galois groups , prépublication ( 1978 ).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.