JOURNAL BROWSE
Search
Advanced SearchSearch Tips
Langlands Functoriality Conjecture
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
  • Journal title : Kyungpook mathematical journal
  • Volume 49, Issue 2,  2009, pp.355-387
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2009.49.2.355
 Title & Authors
Langlands Functoriality Conjecture
Yang, Jae-Hyun;
  PDF(new window)
 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.
 Keywords
automorphic representations;automorphic L-functions;local Langlands conjectures;global Langlands conjecture;Langlands functoriality conjecture;converse theorems;
 Language
English
 Cited by
 References
1.
J. Adams, D. Barbasch abd D. Vogan, The Langlands Classification and Irreducible Characters for real Reductive Groups, Progr. Math. 104, Birkhauser, Boston, 1992.

2.
J. Adams and D. Vogan, L-groups, projective representations, and the Langlands classification, Amer. J. Math., 114(1992), 45-138. crossref(new window)

3.
J. Arthur, The principle of functoriality, Bull. Amer. Math. Soc., 40(1)(2003), 39-53.

4.
J. Arthur, An Introduction to the Trace Formula, In Harmonic Analysis, the Trace Formula, and Shimura Varieties, Proceedings of the Clay Mathematics Institute, 2003 Summer School, The Fields institute, Toronto, Canada, June 2-27, 2003, edited by J. Arthur, D. Ellwood and R. Kottwitz, Amer. Math. Soc. and CMI, 2005, 1-263.

5.
J. Arthur, A notes on L-packets, Pure and Applied Mathematics Quarterly, 2(2006), 199-217. crossref(new window)

6.
J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Studies 120, Princeton Univ. Press, N.J., 1989.

7.
M. Asgari and F. Shahidi, Generic transfer for general spin groups, Duke Math. J., 132(2006), 137-190. crossref(new window)

8.
M. Asgari and F. Shahidi, Generic transfer for GSp(4) to GL(4), Compositio Math., 142(2006), 541-550. crossref(new window)

9.
A. Borel, Automorphic L-functions, Proc. Sympos. Pure Math., 33, Part 2, Amer. Math. Soc., 1979, 27-62.

10.
K. Buzzard, M. Dickinson, N. Shepard-Barron and R. Taylor, On Icosaheral Artin representations, Duke Math. J., 109(2001), 283-318. crossref(new window)

11.
L. Clozel, M. Harris and R. Taylor, Automorphy for some $\ell$-adic lifts of automorphic mod $\ell$ representations (preprint), http://www.math.harvard.edu/-rtaylor/.

12.
J. Cogdell, Lectures on L-functions, Converse Theorems, and Functoriality for GL(n), in Lectures in Automorphic L-functions, Fields institute Monographs, Amer. Math. Soc., 2004, 3-96.

13.
J. Cogdell, H. Kim, I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to $GL_N$, Publ. Math. Inst. Hautes Etudes Sci., 93(2001), 5-30. crossref(new window)

14.
J. Cogdell, H. Kim, I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the classical groups, Publ. Math. Inst. Hautes Etudes Sci., 99(2004), 163-233. crossref(new window)

15.
J. Cogdell and I. Piatetski-Shapiro, Converse theorems for $GL_n$, Publ. Math. Inst. Hautes Etudes Sci., 79(1994), 157-204. crossref(new window)

16.
J. Cogdell and I. Piatetski-Shapiro, Converse theorems for $GL_n$, II, J. reine angew. Math., 507(1999), 165-188.

17.
P. Deligne, Formes modulaires et representations $\ell$-adiques: Sem. Bourbaki 1968/9, no. 355.

18.
P. Deligne, Formes modulaires et representations l-adiques: Lecture Notes in Math. 179, Springer-Verlag, Berlin-Heidelberg-New York, 1971, 139-172.

19.
P. Deligne, Formes modulaires et representations of GL(2), Antwerp II, in Modular Forms of One Variable II, Lecture Notes in Math. 349, Springer-Verlag, New York, 1973, 55-106.

20.
P. Deligne, Les constants des equation fontionnelle des fonctions L, in Modular Forms of One Variable II, Lecture Notes in Math., 349, Springer-Verlag, New York, 1973, 501-595.

21.
V. Drinfeld, Proof of the Petersson conjecture for GL(2) over a global field of characteristic p, Funct. Anal. and its Appl., 22(1988), 28-43. crossref(new window)

22.
V. Drinfeld, Cohomology of compactified modules of F-sheaves of rank 2, J. Soviet Math., 46(1989), 1789-1821. crossref(new window)

23.
S. Gelbart and H. Jacquet, A relation between automorphic representation of GL(2) and GL(3), Ann. Scient. Ec. Normale Sup., 11(1978), 471-552.

24.
S. Gelbart and F. Shahidi, Analytic Properties of Automorphic L-Functions, Perspective in Mathematics, 6(1988), Academic Press.

25.
S. Gelbart, I. Piatetski-Shapiro and S. Rallis, Explicit Constructions of Automorphic L-Functions, Lecture Notes in Math. 1254, Springer-Verlag, New York, 1987.

26.
D. Ginzburg, S. Rallis and D. Soudry, Generic automorphic forms on SO(2n + 1): functorial lift on GL(2n), endoscopy, and the base change, Internat. Math. Res. Notices, no. 14(2001), 729-764.

27.
H. Hamburger, Uber die Riemannsche Funktionalgleichung der $\zeta$-Funktion, Math. Zeit., 10(1921), 240-254. crossref(new window)

28.
H. Hamburger, Uber die Riemannsche Funktionalgleichung der $\zeta$-Funktion, Math. Zeit., 11(1922), 224-245.

29.
H. Hamburger, Uber die Riemannsche Funktionalgleichung der $\zeta$-Funktion, Math. Zeit., 13(1922), 283-311. crossref(new window)

30.
M. Harris, N. Shepherd-Barron and R. Taylor, Ihara's lemma and potential automorphy (preprint), http://www.math.harvard.edu/-rtaylor/.

31.
M. Harris and R. Taylor, On the Geonmetry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton, N. J., 2001.

32.
E. Hecke, Uber die Bestimmung Dirichletscher Reihnen durch Funktionalgleichungen, Math. Ann., 112(1936), 664-699. crossref(new window)

33.
G. Henniart, Caracterisation de la correspondance de Langlands locale par les facteurs $\varepsilon$ de paires, Invent. Math., 113(1993), 339-350. crossref(new window)

34.
G. Henniart, Une preuve simple des conjectures de Langlands de GL(n) sur un corps p-adique, Invent. Math., 139(2000), 439-455. crossref(new window)

35.
T. Ikeda, On the location of poles of the triple L-functions, Compositio Math., 83(1992), 187-237.

36.
H. Jacquet and R. Langlands, Automorphic Forms on GL(2), Lecture Notes in Math. 114, Springer-Verlag, New York, 1970.

37.
H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Automorphic forms on GL(3), I and II, Ann. of Math., 109(1979), 169-258. crossref(new window)

38.
H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Rankin-Selberg convolutions, Amer. J. Math., 105(1983), 367-464. crossref(new window)

39.
H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations I, Amer. J. Math., 103(1981), 499-558. crossref(new window)

40.
H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations II, Amer. J. Math., 103(1981), 777-815. crossref(new window)

41.
D. Jiang and D. Soudry, The local converse theorem for SO(2n+1) and applications, Ann. Math., 157(2003), 743-806. crossref(new window)

42.
D. Jiang and D. Soudry, Generic representations and local langlands reciprocity law for p-adic $SO_{2n+1}$, in Contributions to Automorphic Forms, Geometry and Number Theory, edited by H. Hida, D. Ramakrishnan and F. Shahidi, the John Hopkins University Press, Baltimore, 2004, 457-519.

43.
D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math., 95(1987), 153-215.

44.
C. Khare, M. Larsen and G. Savin, Functoriality and the inverse Galois problem, Compositio Math., 144(2008), 541-564.

45.
C. Khare, M. Larsen and G. Savin, Functoriality and the inverse Galois problem II: Groups of type $B_n$ and $G_2$, arXiv:0807.0861v1, 2008.

46.
H. Kim, Functoriality for the exterior square of $GL_4$ and the symmetric fourth of $GL_2$, J. of Amer. Math. Soc., 16(1)(2002), 139-183.

47.
H. Kim, Automorphic L-functions, in Lectures in Automorphic L-functions, Fields institute Monographs, Amer. Math. Soc., 2004, 97-201.

48.
H. Kim and F. Shahidi, Symmetric cube L-functions for $GL_2$ are entire, Ann. of Math., 150(1999), 645-662. crossref(new window)

49.
H. Kim and F. Shahidi, Functional products for $GL_2{\times}GL_3$ and functorial symmetric cube for $GL_2$, Ann. of Math., 155(2002), 837-893. crossref(new window)

50.
H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J., 112(2002), 177-197. crossref(new window)

51.
A. Knapp and G. Zuckerman, Normalizing factors and L-groups, Proc. Symp. Pure Math., Amer. Math. Soc., Providence, RI, 33, Part 1, 1979, 93-106.

52.
A. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math., 116(1982), 389-455. crossref(new window)

53.
R. Kottwitz and D. Shelstad, Foundations of Twisted Endoscopy, Asterisque, 255(1999).

54.
J. -P. Labesse and R. Langlands, L-indistinguishabilty for SL(2), Canad. J. Math., 31(1979), 726-785. crossref(new window)

55.
L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math., 147(2002), 1-241. crossref(new window)

56.
K. F. Lai, On the Tamagawa number of quasi-split groups, Thesis, Yale University, 1974.

57.
R. Langlands, Problems in the theory of automorphic forms, in Lectures in modern analysis and Applications, Lecture Notes in Math. 170, Springer-Verlag, New York, 1970, 18-61.

58.
R. Langlands, Euler Products, Yale University Press, New Haven 310, 1971.

59.
R. Langlands, Modular forms and $\ell$-adic representations, in Modular Forms of One Variable II, Lecture Notes in Math. 349, Springer-Verlag, New York, 1973, 361-500.

60.
R. Langlands, On Artin's L-functions, Rice University Studies 56, 23-28.

61.
R. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math. 544, Springer-Verlag, New York, 1976.

62.
R. Langlands, On the notion of automorphic representations. A supplement to the preceding paper, Proc. Sympos. Pure Math. 33, Part 1, Amer. Math. Soc., 1979, 203-208.

63.
R. Langlands, Automorphic representations, Shimura varieties, and motives, Proc. Sympos. Pure Math., 33, Part 2, Amer. Math. Soc., 1979, 205-246.

64.
R. Langlands, Base Change for GL(2), Ann. of Math. Studies 96, Princeton Univ. Press, Princeton, N. J. 1980.

65.
R. Langlands, Les debuts d'une formule des traces stables, Publ. Math. Ubiv. Paris VII., 13(1983).

66.
R. Langlands, On the classiciation of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, AMS Mathematical Surveys and Monographs, 31(1989), 101-170. crossref(new window)

67.
R. Langlands, Representations of abelian algebraic groups, Pacific J. Math., 1997, 231-250.

68.
R. Langlands, Where stands functoriality today?, in Representation Theory and Automorphic Forms (March 1996, Edinburgh, Scotland), edited by T. N. Bailey and A. W. Knapp, 1997, 457-471.

69.
R. Langlands, Beyond endoscopy, in Contributions to Automorphic Forms, Geometry and Number Theory, edited by H. Hida, D. Ramakrishnan and F. Shahidi, the John Hopkins University Press, Baltimore 2004, 611-698.

70.
G. Laumon, M. Rapoport and U. Stuhler, D-eliptic sheaves and the Langlands correspondence, Invent. Math., 113(1993), 217-338. crossref(new window)

71.
H. Maass, Uber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihnen durch Funktionalgleichungen, Math. Ann., 121(1949), 141-183. crossref(new window)

72.
B. Mazur, Finding Meaning in Error Terms, Bull. Amer. Math. Soc., 45(2)April(2008), 185-228. crossref(new window)

73.
D. Ramakrishnan, Pure motives and automorphic forms, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., 1994, 377-400.

74.
D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math., 152(2000), 45-111. crossref(new window)

75.
D. Ramakrishnan, Modularity of solvable Artin representations of GO(4)-type, Int. Math. Research Notices, no. 1 (2002), 1-54.

76.
J.-P. Serre, Abelian $\ell$-adic Representations and Elliptic Curves, Benjamin, New York, 1968.

77.
J.-P. Serre, Proprietes conjecturales des groupes de Galois motiviques et des representations $\ell$-adiques, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., 1994, 377-400.

78.
F. Shahidi, Functional equations satisfied by certain L-functions, Compositio Math., 37(1978), 171-208.

79.
F. Shahidi, On certain L-functions, Amer. J. Math., 103(1981), 297-356. crossref(new window)

80.
F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math., 127(1988), 547-584. crossref(new window)

81.
F. Shahidi, A proof of Langlands' conjecture on Plancherel measures; Complementary series for p-adic groups, Ann. of Math., 132(1988), 273-330.

82.
F. Shahidi, Local coefficients as Mellin transforms of Bessel functions; towards a general stability, IMRN, 39(2002), 2075-2119.

83.
F. Shahidi, Third symmetric powe L-functions for GL(2), Compositio Math., 70(1989), 245-273.

84.
F. Shahidi, Langlands Functoriality Conjecture and Number Theory, In Representation Theory and Automorphic Forms, edited by T. Kobayashi, W. Schmid and J. -H. Yang, Progress in Mathematics, Birkhaser, 255(2008), 151-173.

85.
J. Tate, Number theoretic background, in Automorphic Forms, representation Theory and L-functions, Proc. Sympos. Pure Math., 33, Part 2, Amer. Math. Soc., 1979, 3-26.

86.
R. Taylor, Automorphy for some $\ell$-adic lifts of automorphic mod $\ell$ representations II (preprint), http://www.math.harvard.edu/-rtaylor/.

87.
J. Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc., 5(1981), 173-175. crossref(new window)

88.
A. Weil, Uber die Bestimmung Dirichletscher Reihnen durch Funktionalgleichungen, Math. Ann., 168(1967), 149-156. crossref(new window)

89.
A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math., 141(1995), 443-551. crossref(new window)