Langlands Functoriality Conjecture

다국어 초록 (Multilingual Abstract)

Functoriality conjecture is one of the central and influential subjects of the present day mathematics. Functoriality is the profound lifting problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository article, I describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which are powerful tools for the establishment of functoriality of some important cases, and survey the interesting results related to functoriality conjecture.

참고문헌 (Reference)

1 R. Langlands, "Where stands functoriality today?" 457-471, 1997

2 G. Henniart, "Une preuve simple des conjectures de Langlands de GL(n) sur un corps p-adique" 139 : 439-455, 2000

3 H. Maass, "Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihnen durch Funktionalgleichungen" 121 : 141-183, 1949

4 H. Hamburger, "Uber die Riemannsche Funktionalgleichung der -Funktion" 13 : 283-311, 1922

5 A. Weil, "Uber die Bestimmung Dirichletscher Reihnen durch Funktionalgleichungen" 168 : 149-156, 1967

6 E. Hecke, "Uber die Bestimmung Dirichletscher Reihnen durch Funktionalgleichungen" 112 : 664-699, 1936

7 F. Shahidi, "Third symmetric powe L-functions for GL(2)" 70 : 245-273, 1989

8 J. Arthur, "The principle of functoriality" 40 (40): 39-53, 2003

9 D. Jiang, "The local converse theorem for SO(2n+1) and applications" 157 : 743-806, 2003

10 J. Adams, "The Langlands Classification and Irreducible Characters for real Reductive Groups" 104 : 1992

11 H. Kim, "Symmetric cube L-functions for GL2 are entire" 150 : 645-662, 1999

12 J. Arthur, "Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula" Princeton Univ. Press 1989

13 R. Langlands, "Representations of abelian algebraic groups" 231-250, 1997

14 H. Jacquet, "Rankin-Selberg convolutions" 105 : 367-464, 1983

15 D. Ramakrishnan, "Pure motives and automorphic forms, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math." 55 (55): 377-400, 1994

16 J.-P. Serre, "Propri´et´es conjecturales des groupes de Galois motiviques et des representations `-adiques, in Motives (Seattle, WA, 1991)" 55 (55): 377-400, 1994

17 V. Drinfeld, "Proof of the Petersson conjecture for GL(2) over a global field of characteristic p" 22 : 28-43, 1988

18 D. Kazhdan, "Proof of the Deligne-Langlands conjecture for Hecke algebras" 95 : 153-215, 1987

19 R. Langlands, "Problems in the theory of automorphic forms, in Lectures in modern analysis and Applications" Springer-Verlag 18-61, 1970

20 R. Langlands, "On the notion of automorphic representations" 33 (33): 203-208, 1979

21 T. Ikeda, "On the location of poles of the triple L-functions" 83 : 187-237, 1992

22 R. Langlands, "On the classiciation of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups" 31 : 101-170, 1989

23 K. F. Lai, "On the Tamagawa number of quasi-split groups" Yale University 1974

24 F. Shahidi, "On the Ramanujan conjecture and finiteness of poles for certain Lfunctions" 127 : 547-584, 1988

25 M. Harris, "On the Geonmetry and Cohomology of Some Simple Shimura Varieties" Princeton Univ. Press 2001

26 R. Langlands, "On the Functional Equations Satisfied by Eisenstein Series" Springer-Verlag 1976

27 J. Cogdell, "On lifting from classical groups to GLN" 93 : 5-30, 2001

28 F. Shahidi, "On certain L-functions" 103 : 297-356, 1981

29 K. Buzzard, "On Icosaheral Artin representations" 109 : 283-318, 2001

30 H. Jacquet, "On Euler products and the classification of automorphic representations II" 103 : 777-815, 1981

31 H. Jacquet, "On Euler products and the classification of automorphic representations I" 103 : 499-558, 1981

32 R. Langlands, "On Artin’s L-functions" 56 : 23-28,

33 J. Tate, "Number theoretic background, in Automorphic Forms, representation Theory and L-functions" 33 (33): 3-26, 1979

34 A. Knapp, "Normalizing factors and L-groups" 33 (33): 93-106, 1979

35 D. Ramakrishnan, "Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)" 152 : 45-111, 2000

36 D. Ramakrishnan, "Modularity of solvable Artin representations of GO(4)-type" (1) : 1-54, 2002

37 R. Langlands, "Modular forms and `-adic representations, in Modular Forms of One Variable II" Springer-Verlag 361-500, 1973

38 A. Wiles, "Modular elliptic curves and Fermat’s Last Theorem" 141 : 443-551, 1995

39 F. Shahidi, "Local coefficients as Mellin transforms of Bessel functions; towards a general stability" 39 : 2075-2119, 2002

40 R. Langlands, "Les d´ebuts d’une formule des traces stables" 13 : 1983

41 P. Deligne, "Les constants des ´equation fontionnelle des fonctions L , in Modular Forms of One Variable II" Springer-Verlag 501-595, 1973

42 J. Cogdell, "Lectures on L-functions, Converse Theorems, and Functoriality for GL(n), in Lectures in Automorphic L-functions" 3-96, 2004

43 F. Shahidi, "Langlands Functoriality Conjecture and Number Theory, In Representation Theory and Automorphic Forms" 255 : 151-173, 2008

44 J. -P. Labesse, "L-indistinguishabilty for SL(2)" 31 : 726-785, 1979

45 J. Adams, "L-groups, projective representations, and the Langlands classification" 114 : 45-138, 1992

46 M. Harris, "Ihara’s lemma and potential automorphy (preprint)"

47 M. Asgari, "Generic transfer for general spin groups" 132 : 137-190, 2006

48 M. Asgari, "Generic transfer for GSp(4) to GL(4)" 142 : 541-550, 2006

49 D. Jiang, "Generic representations and local langlands reciprocity law for p-adic SO2n+1, in Contributions to Automorphic Forms, Geometry and Number Theory" the John Hopkins University Press 457-519, 2004

50 D. Ginzburg, "Generic automorphic forms on SO(2n + 1): functorial lift on GL(2n), endoscopy, and the base change" 14 : 729-764, 2001

51 H. Kim, "Functoriality for the exterior square of GL4 and the symmetric fourth of GL2" 16 (16): 139-183, 2002

52 J. Cogdell, "Functoriality for the classical groups" 99 : 163-233, 2004

53 C. Khare, "Functoriality and the inverse Galois problem II: Groups of type Bn and G2, arXiv:0807.0861v1"

54 C. Khare, "Functoriality and the inverse Galois problem" 144 : 541-564, 2008

55 H. Kim, "Functional products for GL2×GL3 and functorial symmetric cube for GL2" 155 : 837-893, 2002

56 F. Shahidi, "Functional equations satisfied by certain L-functions" 37 : 171-208, 1978

57 R. Kottwitz, "Foundations of Twisted Endoscopy" 255 : 1999

58 P. Deligne, "Formes modulaires et repr´esentations of GL(2), Antwerp II, in Modular Forms of One Variable II" Springer-Verlag 55-106, 1973

59 P. Deligne, "Formes modulaires et repr´esentations `-adiques: S´em. Bourbaki 1968/9, no. 355" Springer-Verlag 139-172, 1971

60 B. Mazur, "Finding Meaning in Error Terms" 45 (45): 185-228, 2008

61 S. Gelbart, "Explicit Constructions of Automorphic L-Functions" Springer-Verlag 1987

62 R. Langlands, "Euler Products" Yale University Press 310-, 1971

63 G. Laumon, "D-eliptic sheaves and the Langlands correspondence" 113 : 217-338, 1993

64 H. Kim, "Cuspidality of symmetric powers with applications" 112 : 177-197, 2002

65 J. Cogdell, "Converse theorems for GLn, II" 507 : 165-188, 1999

66 J. Cogdell, "Converse theorems for GLn" 79 : 157-204, 1994

67 V. Drinfeld, "Cohomology of compactified modules of F-sheaves of rank 2" 46 : 1789-1821, 1989

68 A. Knapp, "Classification of irreducible tempered representations of semisimple groups" 116 : 389-455, 1982

69 L. Lafforgue, "Chtoucas de Drinfeld et correspondance de Langlands" 147 : 1-241, 2002

70 G. Henniart, "Caract´erisation de la correspondance de Langlands locale par les facteurs " de paires" 113 : 339-350, 1993

71 R. Langlands, "Beyond endoscopy, in Contributions to Automorphic Forms, Geometry and Number Theory" John Hopkins University Press 611-698, 2004

72 R. Langlands, "Base Change for GL(2)" Princeton Univ. Press 1980

73 R. Taylor, "Automorphy for some `-adic lifts of automorphic mod ` representations II (preprint)"

74 L. Clozel, "Automorphy for some `-adic lifts of automorphic mod ` representations (preprint)"

75 R. Langlands, "Automorphic representations, Shimura varieties, and motives" 33 (33): 205-246, 1979

76 H. Jacquet, "Automorphic forms on GL(3), I and II" 109 : 169-258, 1979

77 H. Kim, "Automorphic L-functions, in Lectures in Automorphic L-functions, Fields institute Monographs" 97-201, 2004

78 A. Borel, "Automorphic L-functions" 33 : 27-62, 1979

79 H. Jacquet, "Automorphic Forms on GL(2)" Springer-Verlag 1970

80 J. Tunnell, "Artin’s conjecture for representations of octahedral type" 5 : 173-175, 1981

81 S. Gelbart, "Analytic Properties of Automorphic L-Functions" Academic Press 1988

82 J. Arthur, "An Introduction to the Trace Formula, In Harmonic Analysis, the Trace Formula, and Shimura Varieties" Amer. Math. Soc. and CMI 2-27, 2003

83 J.-P. Serre, "Abelian `-adic Representations and Elliptic Curves" Benjamin 1968

84 S. Gelbart, "A relation between automorphic representation of GL(2) and GL(3)" 11 : 471-552, 1978

85 F. Shahidi, "A proof of Langlands’ conjecture on Plancherel measures; Complementary series for p-adic groups" 132 : 273-330, 1988

86 J. Arthur, "A notes on L-packets" 2 : 199-217, 2006