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Local Langlands conjectures

January 01, 1970

In mathematics, the local Langlands conjectures, introduced by Robert Langlands (1967, 1970), are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.

Local Langlands conjectures for GL1 edit

The local Langlands conjectures for GL1(K) follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1(K)= K* to the abelianization of the Weil group. In particular irreducible smooth representations of GL1(K) are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1(C). This gives the Langlands correspondence between homomorphisms of the Weil group to GL1(C) and irreducible smooth representations of GL1(K).

Representations of the Weil group edit

Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw−1=||w||N, or equivalently a representation of the Weil–Deligne group. In addition the representation of the Weil group should have an open kernel, and should be (Frobenius) semisimple.

For every Frobenius semisimple complex n-dimensional Weil–Deligne representation ρ of the Weil group of F there is an L-function L(s,ρ) and a local ε-factor ε(s,ρ,ψ) (depending on a character ψ of F).

Representations of GLn(F) edit

The representations of GLn(F) appearing in the local Langlands correspondence are smooth irreducible complex representations.

  • "Smooth" means that every vector is fixed by some open subgroup.
  • "Irreducible" means that the representation is nonzero and has no subrepresentations other than 0 and itself.

Smooth irreducible complex representations are automatically admissible.

The Bernstein–Zelevinsky classification reduces the classification of irreducible smooth representations to cuspidal representations.

For every irreducible admissible complex representation π there is an L-function L(s,π) and a local ε-factor ε(s,π,ψ) (depending on a character ψ of F). More generally, if there are two irreducible admissible representations π and π' of general linear groups there are local Rankin–Selberg convolution L-functions L(s,π×π') and ε-factors ε(s,π×π',ψ).

Bushnell & Kutzko (1993) described the irreducible admissible representations of general linear groups over local fields.

Local Langlands conjectures for GL2 edit

The local Langlands conjecture for GL2 of a local field says that there is a (unique) bijection π from 2-dimensional semisimple Weil-Deligne representations of the Weil group to irreducible smooth representations of GL2(F) that preserves L-functions, ε-factors, and commutes with twisting by characters of F*.

Jacquet & Langlands (1970) verified the local Langlands conjectures for GL2 in the case when the residue field does not have characteristic 2. In this case the representations of the Weil group are all of cyclic or dihedral type. Gelfand & Graev (1962) classified the smooth irreducible representations of GL2(F) when F has odd residue characteristic (see also (Gelfand, Graev & Pyatetskii-Shapiro 1969, chapter 2)), and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case. Weil (1974) pointed out that when the residue field has characteristic 2, there are some extra exceptional 2-dimensional representations of the Weil group whose image in PGL2(C) is of tetrahedral or octahedral type. (For global Langlands conjectures, 2-dimensional representations can also be of icosahedral type, but this cannot happen in the local case as the Galois groups are solvable.) Tunnell (1978) proved the local Langlands conjectures for the general linear group GL2(K) over the 2-adic numbers, and over local fields containing a cube root of unity. Kutzko (1980, 1980b) proved the local Langlands conjectures for the general linear group GL2(K) over all local fields.

Cartier (1981) and Bushnell & Henniart (2006) gave expositions of the proof.

Local Langlands conjectures for GLn edit

The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρπ from equivalence classes of irreducible admissible representations π of GLn(F) to equivalence classes of continuous Frobenius semisimple complex n-dimensional Weil–Deligne representations ρπ of the Weil group of F, that preserve L-functions and ε-factors of pairs of representations, and coincide with the Artin map for 1-dimensional representations. In other words,

  • L(s,ρπ⊗ρπ') = L(s,π×π')
  • ε(s,ρπ⊗ρπ',ψ) = ε(s,π×π',ψ)

Laumon, Rapoport & Stuhler (1993) proved the local Langlands conjectures for the general linear group GLn(K) for positive characteristic local fields K. Carayol (1992) gave an exposition of their work.

Harris & Taylor (2001) proved the local Langlands conjectures for the general linear group GLn(K) for characteristic 0 local fields K. Henniart (2000) gave another proof. Carayol (2000) and Wedhorn (2008) gave expositions of their work.

Local Langlands conjectures for other groups edit

Borel (1979) and Vogan (1993) discuss the Langlands conjectures for more general groups. The Langlands conjectures for arbitrary reductive groups G are more complicated to state than the ones for general linear groups, and it is unclear what the best way of stating them should be. Roughly speaking, admissible representations of a reductive group are grouped into disjoint finite sets called L-packets, which should correspond to some classes of homomorphisms, called L-parameters, from the local Langlands group to the L-group of G. Some earlier versions used the Weil−Deligne group or the Weil group instead of the local Langlands group, which gives a slightly weaker form of the conjecture.

Langlands (1989) proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible admissible representations (up to infinitesimal equivalence), or, equivalently, of their irreducible  -modules.

Gan & Takeda (2011) proved the local Langlands conjectures for the symplectic similitude group GSp(4) and used that in Gan & Takeda (2010) to deduce it for the symplectic group Sp(4).

References edit

  • Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, vol. XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, ISBN 978-0-8218-1437-6, MR 0546608
  • Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
  • Bushnell, Colin J.; Kutzko, Philip C. (1993), The admissible dual of GL(N) via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, ISBN 978-0-691-03256-6, MR 1204652
  • Carayol, Henri (1992), "Variétés de Drinfeld compactes, d'après Laumon, Rapoport et Stuhler", Astérisque, 206: 369–409, ISSN 0303-1179, MR 1206074
  • Carayol, Henri (2000), "Preuve de la conjecture de Langlands locale pour GLn: travaux de Harris-Taylor et Henniart", Séminaire Bourbaki. Vol. 1998/99., Astérisque, 266: 191–243, ISSN 0303-1179, MR 1772675
  • Cartier, Pierre (1981), "La conjecture locale de Langlands pour GL(2) et la démonstration de Ph. Kutzko", Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math. (in French), vol. 842, Berlin, New York: Springer-Verlag, pp. 112–138, doi:10.1007/BFb0089931, ISBN 978-3-540-10292-2, MR 0636520
  • Gan, Wee Teck; Takeda, Shuichiro (2010), "The local Langlands conjecture for Sp(4)", International Mathematics Research Notices, 2010 (15): 2987–3038, arXiv:0805.2731, doi:10.1093/imrn/rnp203, ISSN 1073-7928, MR 2673717, S2CID 5990821
  • Gan, Wee Teck; Takeda, Shuichiro (2011), "The local Langlands conjecture for GSp(4)", Annals of Mathematics, 173 (3): 1841–1882, arXiv:0706.0952, doi:10.4007/annals.2011.173.3.12, S2CID 5990821
  • Gelfand, I. M.; Graev, M. I. (1962), "Construction of irreducible representations of simple algebraic groups over a finite field", Doklady Akademii Nauk SSSR, 147: 529–532, ISSN 0002-3264, MR 0148765 English translation in volume 2 of Gelfand's collected works.
  • Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1969) [1966], Representation theory and automorphic functions, Generalized functions, vol. 6, Philadelphia, Pa.: W. B. Saunders Co., ISBN 978-0-12-279506-0, MR 0220673
  • Harris, Michael; Taylor, Richard (2001), The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, ISBN 978-0-691-09090-0, MR 1876802
  • Henniart, Guy (2000), "Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique", Inventiones Mathematicae, 139 (2): 439–455, Bibcode:2000InMat.139..439H, doi:10.1007/s002220050012, ISSN 0020-9910, MR 1738446, S2CID 120799103
  • Henniart, Guy (2006), "On the local Langlands and Jacquet-Langlands correspondences", in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; et al. (eds.), International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 1171–1182, ISBN 978-3-03719-022-7, MR 2275640
  • Jacquet, Hervé; Langlands, Robert P. (1970), Automorphic Forms on GL (2), Lecture Notes in Mathematics, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654, S2CID 122773458
  • Kudla, Stephen S. (1994), "The local Langlands correspondence: the non-Archimedean case", in Jannsen, Uwe; Kleiman, Steven; Serre, Jean-Pierre (eds.), Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Providence, R.I.: American Mathematical Society, pp. 365–391, ISBN 978-0-8218-1637-0, MR 1265559
  • Kutzko, Philip (1980), "The Langlands conjecture for GL2 of a local field", Bulletin of the American Mathematical Society, New Series, 2 (3): 455–458, doi:10.1090/S0273-0979-1980-14765-5, ISSN 0002-9904, MR 0561532
  • Kutzko, Philip (1980b), "The Langlands conjecture for Gl2 of a local field", Annals of Mathematics, Second Series, 112 (2): 381–412, doi:10.2307/1971151, ISSN 0003-486X, JSTOR 1971151, MR 0592296
  • Langlands, Robert (1967), Letter to Prof. Weil
  • Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, vol. 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614
  • Langlands, Robert P. (1989) [1973], "On the classification of irreducible representations of real algebraic groups", in Sally, Paul J.; Vogan, David A. (eds.), Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Providence, R.I.: American Mathematical Society, pp. 101–170, ISBN 978-0-8218-1526-7, MR 1011897
  • Laumon, G.; Rapoport, M.; Stuhler, U. (1993), "D-elliptic sheaves and the Langlands correspondence", Inventiones Mathematicae, 113 (2): 217–338, Bibcode:1993InMat.113..217L, doi:10.1007/BF01244308, ISSN 0020-9910, MR 1228127, S2CID 124557672
  • Tunnell, Jerrold B. (1978), "On the local Langlands conjecture for GL(2)", Inventiones Mathematicae, 46 (2): 179–200, Bibcode:1978InMat..46..179T, doi:10.1007/BF01393255, ISSN 0020-9910, MR 0476703, S2CID 117747963
  • Vogan, David A. (1993), "The local Langlands conjecture", in Adams, Jeffrey; Herb, Rebecca; Kudla, Stephen; Li, Jian-Shu; Lipsman, Ron; Rosenberg, Jonathan (eds.), Representation theory of groups and algebras, Contemp. Math., vol. 145, Providence, R.I.: American Mathematical Society, pp. 305–379, ISBN 978-0-8218-5168-5, MR 1216197
  • Wedhorn, Torsten (2008), (PDF), in Göttsche, Lothar; Harder, G.; Raghunathan, M. S. (eds.), School on Automorphic Forms on GL(n), ICTP Lect. Notes, vol. 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp. 237–320, arXiv:math/0011210, Bibcode:2000math.....11210W, ISBN 978-92-95003-37-8, MR 2508771, archived from the original (PDF) on 2020-05-07
  • Weil, André (1974), "Exercices dyadiques", Inventiones Mathematicae, 27 (1–2): 1–22, Bibcode:1974InMat..27....1W, doi:10.1007/BF01389962, ISSN 0020-9910, MR 0379445, S2CID 189830448

External links edit

  • Harris, Michael (2000), The local Langlands correspondence (PDF), Notes of (half) a course at the IHP
  • The work of Robert Langlands
  • Automorphic Forms - The local Langlands conjecture Lecture by Richard Taylor

January 01, 1970
local, langlands, conjectures, mathematics, local, langlands, conjectures, introduced, robert, langlands, 1967, 1970, part, langlands, program, they, describe, correspondence, between, complex, representations, reductive, algebraic, group, over, local, field, . In mathematics the local Langlands conjectures introduced by Robert Langlands 1967 1970 are part of the Langlands program They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F and representations of the Langlands group of F into the L group of G This correspondence is not a bijection in general The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non abelian Galois groups Contents 1 Local Langlands conjectures for GL1 2 Representations of the Weil group 3 Representations of GLn F 4 Local Langlands conjectures for GL2 5 Local Langlands conjectures for GLn 6 Local Langlands conjectures for other groups 7 References 8 External linksLocal Langlands conjectures for GL1 editThe local Langlands conjectures for GL1 K follow from and are essentially equivalent to local class field theory More precisely the Artin map gives an isomorphism from the group GL1 K K to the abelianization of the Weil group In particular irreducible smooth representations of GL1 K are 1 dimensional as the group is abelian so can be identified with homomorphisms of the Weil group to GL1 C This gives the Langlands correspondence between homomorphisms of the Weil group to GL1 C and irreducible smooth representations of GL1 K Representations of the Weil group editRepresentations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups To get a bijection one has to slightly modify the notion of a representation of the Weil group to something called a Weil Deligne representation This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw 1 w N or equivalently a representation of the Weil Deligne group In addition the representation of the Weil group should have an open kernel and should be Frobenius semisimple For every Frobenius semisimple complex n dimensional Weil Deligne representation r of the Weil group of F there is an L function L s r and a local e factor e s r ps depending on a character ps of F Representations of GLn F editThe representations of GLn F appearing in the local Langlands correspondence are smooth irreducible complex representations Smooth means that every vector is fixed by some open subgroup Irreducible means that the representation is nonzero and has no subrepresentations other than 0 and itself Smooth irreducible complex representations are automatically admissible The Bernstein Zelevinsky classification reduces the classification of irreducible smooth representations to cuspidal representations For every irreducible admissible complex representation p there is an L function L s p and a local e factor e s p ps depending on a character ps of F More generally if there are two irreducible admissible representations p and p of general linear groups there are local Rankin Selberg convolution L functions L s p p and e factors e s p p ps Bushnell amp Kutzko 1993 described the irreducible admissible representations of general linear groups over local fields Local Langlands conjectures for GL2 editThe local Langlands conjecture for GL2 of a local field says that there is a unique bijection p from 2 dimensional semisimple Weil Deligne representations of the Weil group to irreducible smooth representations of GL2 F that preserves L functions e factors and commutes with twisting by characters of F Jacquet amp Langlands 1970 verified the local Langlands conjectures for GL2 in the case when the residue field does not have characteristic 2 In this case the representations of the Weil group are all of cyclic or dihedral type Gelfand amp Graev 1962 classified the smooth irreducible representations of GL2 F when F has odd residue characteristic see also Gelfand Graev amp Pyatetskii Shapiro 1969 chapter 2 and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case Weil 1974 pointed out that when the residue field has characteristic 2 there are some extra exceptional 2 dimensional representations of the Weil group whose image in PGL2 C is of tetrahedral or octahedral type For global Langlands conjectures 2 dimensional representations can also be of icosahedral type but this cannot happen in the local case as the Galois groups are solvable Tunnell 1978 proved the local Langlands conjectures for the general linear group GL2 K over the 2 adic numbers and over local fields containing a cube root of unity Kutzko 1980 1980b proved the local Langlands conjectures for the general linear group GL2 K over all local fields Cartier 1981 and Bushnell amp Henniart 2006 gave expositions of the proof Local Langlands conjectures for GLn editThe local Langlands conjectures for general linear groups state that there are unique bijections p rp from equivalence classes of irreducible admissible representations p of GLn F to equivalence classes of continuous Frobenius semisimple complex n dimensional Weil Deligne representations rp of the Weil group of F that preserve L functions and e factors of pairs of representations and coincide with the Artin map for 1 dimensional representations In other words L s rp rp L s p p e s rp rp ps e s p p ps Laumon Rapoport amp Stuhler 1993 proved the local Langlands conjectures for the general linear group GLn K for positive characteristic local fields K Carayol 1992 gave an exposition of their work Harris amp Taylor 2001 proved the local Langlands conjectures for the general linear group GLn K for characteristic 0 local fields K Henniart 2000 gave another proof Carayol 2000 and Wedhorn 2008 gave expositions of their work Local Langlands conjectures for other groups editBorel 1979 and Vogan 1993 discuss the Langlands conjectures for more general groups The Langlands conjectures for arbitrary reductive groups G are more complicated to state than the ones for general linear groups and it is unclear what the best way of stating them should be Roughly speaking admissible representations of a reductive group are grouped into disjoint finite sets called L packets which should correspond to some classes of homomorphisms called L parameters from the local Langlands group to the L group of G Some earlier versions used the Weil Deligne group or the Weil group instead of the local Langlands group which gives a slightly weaker form of the conjecture Langlands 1989 proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible admissible representations up to infinitesimal equivalence or equivalently of their irreducible g K displaystyle mathfrak g K nbsp modules Gan amp Takeda 2011 proved the local Langlands conjectures for the symplectic similitude group GSp 4 and used that in Gan amp Takeda 2010 to deduce it for the symplectic group Sp 4 References editBorel Armand 1979 Automorphic L functions in Borel Armand Casselman W eds Automorphic forms representations and L functions Proc Sympos Pure Math Oregon State Univ Corvallis Ore 1977 Part 2 vol XXXIII Providence R I American Mathematical Society pp 27 61 ISBN 978 0 8218 1437 6 MR 0546608 Bushnell Colin J Henniart Guy 2006 The local Langlands conjecture for GL 2 Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences vol 335 Berlin New York Springer Verlag doi 10 1007 3 540 31511 X ISBN 978 3 540 31486 8 MR 2234120 Bushnell Colin J Kutzko Philip C 1993 The admissible dual of GL N via compact open subgroups Annals of Mathematics Studies vol 129 Princeton University Press ISBN 978 0 691 03256 6 MR 1204652 Carayol Henri 1992 Varietes de Drinfeld compactes d apres Laumon Rapoport et Stuhler Asterisque 206 369 409 ISSN 0303 1179 MR 1206074 Carayol Henri 2000 Preuve de la conjecture de Langlands locale pour GLn travaux de Harris Taylor et Henniart Seminaire Bourbaki Vol 1998 99 Asterisque 266 191 243 ISSN 0303 1179 MR 1772675 Cartier Pierre 1981 La conjecture locale de Langlands pour GL 2 et la demonstration de Ph Kutzko Bourbaki Seminar Vol 1979 80 Lecture Notes in Math in French vol 842 Berlin New York Springer Verlag pp 112 138 doi 10 1007 BFb0089931 ISBN 978 3 540 10292 2 MR 0636520 Gan Wee Teck Takeda Shuichiro 2010 The local Langlands conjecture for Sp 4 International Mathematics Research Notices 2010 15 2987 3038 arXiv 0805 2731 doi 10 1093 imrn rnp203 ISSN 1073 7928 MR 2673717 S2CID 5990821 Gan Wee Teck Takeda Shuichiro 2011 The local Langlands conjecture for GSp 4 Annals of Mathematics 173 3 1841 1882 arXiv 0706 0952 doi 10 4007 annals 2011 173 3 12 S2CID 5990821 Gelfand I M Graev M I 1962 Construction of irreducible representations of simple algebraic groups over a finite field Doklady Akademii Nauk SSSR 147 529 532 ISSN 0002 3264 MR 0148765 English translation in volume 2 of Gelfand s collected works Gelfand I M Graev M I Pyatetskii Shapiro I I 1969 1966 Representation theory and automorphic functions Generalized functions vol 6 Philadelphia Pa W B Saunders Co ISBN 978 0 12 279506 0 MR 0220673 Harris Michael Taylor Richard 2001 The geometry and cohomology of some simple Shimura varieties Annals of Mathematics Studies vol 151 Princeton University Press ISBN 978 0 691 09090 0 MR 1876802 Henniart Guy 2000 Une preuve simple des conjectures de Langlands pour GL n sur un corps p adique Inventiones Mathematicae 139 2 439 455 Bibcode 2000InMat 139 439H doi 10 1007 s002220050012 ISSN 0020 9910 MR 1738446 S2CID 120799103 Henniart Guy 2006 On the local Langlands and Jacquet Langlands correspondences in Sanz Sole Marta Soria Javier Varona Juan Luis et al eds International Congress of Mathematicians Vol II Eur Math Soc Zurich pp 1171 1182 ISBN 978 3 03719 022 7 MR 2275640 Jacquet Herve Langlands Robert P 1970 Automorphic Forms on GL 2 Lecture Notes in Mathematics vol 114 Berlin New York Springer Verlag doi 10 1007 BFb0058988 ISBN 978 3 540 04903 6 MR 0401654 S2CID 122773458 Kudla Stephen S 1994 The local Langlands correspondence the non Archimedean case in Jannsen Uwe Kleiman Steven Serre Jean Pierre eds Motives Seattle WA 1991 Proc Sympos Pure Math vol 55 Providence R I American Mathematical Society pp 365 391 ISBN 978 0 8218 1637 0 MR 1265559 Kutzko Philip 1980 The Langlands conjecture for GL2 of a local field Bulletin of the American Mathematical Society New Series 2 3 455 458 doi 10 1090 S0273 0979 1980 14765 5 ISSN 0002 9904 MR 0561532 Kutzko Philip 1980b The Langlands conjecture for Gl2 of a local field Annals of Mathematics Second Series 112 2 381 412 doi 10 2307 1971151 ISSN 0003 486X JSTOR 1971151 MR 0592296 Langlands Robert 1967 Letter to Prof Weil Langlands R P 1970 Problems in the theory of automorphic forms Lectures in modern analysis and applications III Lecture Notes in Math vol 170 Berlin New York Springer Verlag pp 18 61 doi 10 1007 BFb0079065 ISBN 978 3 540 05284 5 MR 0302614 Langlands Robert P 1989 1973 On the classification of irreducible representations of real algebraic groups in Sally Paul J Vogan David A eds Representation theory and harmonic analysis on semisimple Lie groups Math Surveys Monogr vol 31 Providence R I American Mathematical Society pp 101 170 ISBN 978 0 8218 1526 7 MR 1011897 Laumon G Rapoport M Stuhler U 1993 D elliptic sheaves and the Langlands correspondence Inventiones Mathematicae 113 2 217 338 Bibcode 1993InMat 113 217L doi 10 1007 BF01244308 ISSN 0020 9910 MR 1228127 S2CID 124557672 Tunnell Jerrold B 1978 On the local Langlands conjecture for GL 2 Inventiones Mathematicae 46 2 179 200 Bibcode 1978InMat 46 179T doi 10 1007 BF01393255 ISSN 0020 9910 MR 0476703 S2CID 117747963 Vogan David A 1993 The local Langlands conjecture in Adams Jeffrey Herb Rebecca Kudla Stephen Li Jian Shu Lipsman Ron Rosenberg Jonathan eds Representation theory of groups and algebras Contemp Math vol 145 Providence R I American Mathematical Society pp 305 379 ISBN 978 0 8218 5168 5 MR 1216197 Wedhorn Torsten 2008 The local Langlands correspondence for GL n over p adic fields PDF in Gottsche Lothar Harder G Raghunathan M S eds School on Automorphic Forms on GL n ICTP Lect Notes vol 21 Abdus Salam Int Cent Theoret Phys Trieste pp 237 320 arXiv math 0011210 Bibcode 2000math 11210W ISBN 978 92 95003 37 8 MR 2508771 archived from the original PDF on 2020 05 07 Weil Andre 1974 Exercices dyadiques Inventiones Mathematicae 27 1 2 1 22 Bibcode 1974InMat 27 1W doi 10 1007 BF01389962 ISSN 0020 9910 MR 0379445 S2CID 189830448External links editHarris Michael 2000 The local Langlands correspondence PDF Notes of half a course at the IHP The work of Robert Langlands Automorphic Forms The local Langlands conjecture Lecture by Richard Taylor Retrieved from https en wikipedia org w index php title Local Langlands conjectures amp oldid 1210688297, wikipedia, wiki, book, books, library,

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