Bulletin (New Series) of the American Mathematical Society

An elementary introduction to the Langlands program

Stephen Gelbart

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Article information

Source
Bull. Amer. Math. Soc. (N.S.) Volume 10, Number 2 (1984), 177-219.

Dates
First available in Project Euclid: 4 July 2007

Permanent link to this document
http://projecteuclid.org/euclid.bams/1183551573

Mathematical Reviews number (MathSciNet)
MR733692

Zentralblatt MATH identifier
0539.12008

Subjects
Primary: 10D40 12A67
Secondary: 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings [See also 20G05]

Citation

Gelbart, Stephen. An elementary introduction to the Langlands program. Bulletin (New Series) of the American Mathematical Society 10 (1984), no. 2, 177--219. http://projecteuclid.org/euclid.bams/1183551573.


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References

  • [And] A. N. Andrianov, Zeta functions and the Siegel modular forms, Lie Groups and Their Representations (I. M. Gelfand, ed.), Wiley, New York, 1975, pp. 9-20.
  • [Art] J. Arthur, Automorphic representations and number theory, Canad. Math. Soc. Conf. Proc. Vol. 1 1981, pp. 3-54.
  • [Art 2] J. Arthur, Eisenstein series and the trace formula, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc. Providence, R. I., 1979, pp. 253-274.
  • [Art 3] J. Arthur, A trace formula for reductive groups. I, II, Duke Math. J. 45 (1978), 911-952; Compositio Math. 40 (1980), 87-121.
  • [Art 4] J. Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1-74.
  • [Asai] T. Asai, On certain Dirichlet series associated with Hubert modular forms and Rankin's method, Math. Ann. 226 (1977), 81-94.
  • [Be Ze] I. N. Bernstein and A. Zelevinskii, Representations of the group GL (n, F) when F is a nonarchimedean local field, Russian Math. Surveys 31 (1976), no. 3, 1-68.
  • [Bi SD] B. J. Birch and H. P. F. Swinnerton-Dyer, Elliptic curves and modular functions of one variable, Lecture Notes in Math., Vol. 476, Springer, New York, 1975, pp. 2-32.
  • [Bo] A. Borel, Automorphic L-functionsProc. Sympos. Pure Math., Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 27-61.
  • [Bo 2] A. Borel, Formes automorphes et series de Dirichlet, Sém. Bourbaki, No. 466, Lecture Notes in Math., Vol. 514, Springer, New York, pp. 189-222.
  • [Bo Ja] A. Borel and H. Jacquet, Automorphic forms and automorphic representations, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 189-202.
  • [Bo Shaf] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, 1967.
  • [Cart l] P. Cartier, La conjecture locale de Langlands pour GL (2) et la demonstration de Ph. Kutzko, Sem. Bourbaki, 1979/80, No. 550, Lecture Notes in Math., Vol. 842, Springer, 1981.
  • [Cart2] P. Cartier, Representations of p-adic groups: A survey, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I. 1979, pp. 111-156.
  • [Cas] W. Casselman, The Hasse-Weil ζ-function of some moduli varieties of dimension greater than one, Proc. Sympos. Pure Math., Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 141-164.
  • [Cas2] W. Casselman, "GLn", Proc. Durham Sympos. Algebraic Number Fields (London), Academic Press, New York, 1977.
  • [Cassels] J. W. Cassels, Rational quadratic forms, Academic Press, London and New York, 1978.
  • [Cas Fro] J. W. Cassels and A. Frölich (Editors), Algebraic number theory, Proc. Instructional Conf. (Brighton), Thompson Book, Washington, D. C, 1967.
  • [Cloz] L. Clozel, Base change for real groups, preprints, 1980-1982.
  • [Coates] J. Coates, The work of Mazur and Wiles on cyclotomic fields, Sém. Bourbaki 1980/81, Lecture Notes in Math., Vol. 575, Springer-Verlag, 1982, pp. 220-242.
  • [Deligne] P. Deligne, Formes modulaires et representations de GL2, Modular Functions of One Variable, Lecture Notes in Math., Vol. 349, Springer, New York, 1973, pp. 55-106.
  • [Deligne 2] P. Deligne, Les constantes des equations fonctionelles des fonctions L, Modular Functions of One Variable. II, Lecture Notes in Math., Vol. 249, Springer-Verlag, 1973, pp. 501-597.
  • [De Se] P. Deligne and J.-P. Serre, Formes modulaires de poids1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507-530; also article in Proc. Durham Sympos. Algebraic Number Fields (London), Academic Press, New York, 1977, pp. 193-268.
  • [Drin] V. G. Drinfeld, Langlands conjecture for GL(2) over function fields, Proc. Internat. Congress. Math., Helsinki, 1978.
  • [Dwork] D. Dwork, On the Artin root number, Amer. J. Math. 78 (1956), 444-472.
  • [Flath] D. Flath, A comparison of the automorphic representations of GL(3) and its twisted forms, Pacific J. Math. 97 (1981), 373-402.
  • [Flick 1] Y. Flicker, The adjoint lifting from SL(2) to PGL(3), I.H.E.S., 1981, preprint.
  • [Flick 2] Y. Flicker, L-packets and liftings for U(3), Princeton, 1982, preprint.
  • [Flick 3] Y. Flicker, Automorphic forms on covering groups of GL2, Invent. Math. 57 (1980), 119-182.
  • [Flick 4] Y. Flicker, The trace formula and base change for GL(3), Lecture Notes in Math., Vol. 927, Springer, New York, 1982.
  • [Ge l] S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Stud., No. 83, Princeton Univ. Press, Princeton, N. J., 1975.
  • [Ge2] S. Gelbart, Elliptic curves and automorphic representations, Adv. in Math. 21 (1976), 235-292.
  • [Ge 3] S. Gelbart, Automorphic forms and Artin's conjecture, Lecture Notes in Math., Vol. 627, Springer-Verlag, 1977.
  • [Ge 4] S. Gelbart, Examples of dual reductive pairs, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 287-296.
  • [Ge Ja] S. Gelbart and H. Jacquet, A relation between automorphic representations of GL (2) and GL(3), Ann. Sci. École. Norm. Sup. (4) 11 (1978), 471-552.
  • [Ge Ja 2] S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 213-251.
  • [Ge Kn] S. Gelbart and A. Knapp, L-indistinguishability and R groups for the special linear groups, Adv. in Math. 43 (1982), 101-121.
  • [Ge PS l] S. Gelbart and I. Piatetski-Shapiro, On Shimura's correspondence for modular forms of half-integral weight, Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Stud. Math., No. 10, Bombay, 1981, pp. 1-39.
  • [Ge PS 2] S. Gelbart and I. Piatetski-Shapiro, Automorphic forms and L-functions for the unitary group, Proc. Maryland Special Year Conf. Lie Groups, 1982-1983, Lecture Notes in Math., Springer-Verlag (to appear).
  • [Ge Fo] I. M. Gelfand and S. Fomin, Geodesic flows on manifolds of constant negative curvature, Uspekhi Mat. Nauk 7 (1952), no. 1, 118-137; English transl., Amer. Math. Soc. Transl. (2) 1 (1965), 49-65.
  • [GG PS] I. M. Gelfand, M. Graev and I. Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, 1969.
  • [Ge Ka] I. M. Gelfand and D. Kazhdan, Representations of GL (n, K) where K is a local field, Lie Groups and their Representations, Wiley, 1975, 95-118.
  • [Ger] P. Gerardin, Cuspidal unramified series for central simple algebras over local fields, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 157-169.
  • [Ger Lab] P. Gerardin and J.-P. Labesse, The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani), Proc. Sympos. Pure Math., Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 115-134.
  • [God 1] R. Godement, Fonctions automorphes et produits euleriens, Sém. Bourbaki 1968/69, No. 349, Lecture Notes in Math., Vol. 179, Springer-Verlag, New York, 1970, pp. 37-53.
  • [God 2] R. Godement, Introduction aux travaux de Selberg, Sem. Bourbaki, 9e année: 1956/1957, No. 144.
  • [Go Ja] R. Godement and H. Jacquet, Zeta-functions of simple algebras, Lecture Notes in Math., Vol. 260, Springer, New York, 1972.
  • [Goldstein] L. Goldstein, Analytic number theory, Prentice-Hall, 1971.
  • [Gross] K. I. Gross, On the evolution of non-commutative harmonic analysis, Amer. Math. Monthly 85 (1978), 525-548.
  • [Gross B] B. Gross, On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication, Number Theory Related to Fermat's Last Theorem, Progress in Math., Vol. 26, Birkhauser, Boston, 1982.
  • [Harder] G. Harder, Chevalley groups over function fields and automorphic forms, Ann. of Math. (2) 100 (1974), 249-306.
  • [HC] Harish-Chandra, Automorphic forms on semi-simple Lie groups, Lecture Notes in Math., Vol. 62, Springer, New York, 1968.
  • [Hen] G. Henniart, La conjecture de Langlands locale pour GL(3), I.H.E.S., 1982, preprint.
  • [Ho 1] R. Howe, θ-series and invariant theory, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I. 1979, pp. 275-286.
  • [Ho 2] R. Howe, Tamely ramified supercuspidal representations of GLn (F), Pacific J. Math. 73 (1977), 437-460.
  • [Ho 3] R. Howe, Some qualitative results on the representation theory of GLn over a p-adic field, Pacific J. Math. 73 (1977), 479-538.
  • [Humph] J. E. Humphreys, Linear algebraic groups, Springer-Verlag, 1975.
  • [Jacquet 1] H. Jacquet, Representations des groupes lineares p-adiques, Theory of Group Representations and Fourier Analysis, (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Montecatini Terme, 1970), Edizioni Cremonese, Rome, 1971, pp. 119-220.
  • [Jacquet 2] H. Jacquet, Automorphic forms on GL2. II, Lecture Notes in Math., Vol. 278, Springer, New York, 1972.
  • [Jacquet 3] H. Jacquet, Principal L-functions of the linear group, Proc. Sympos. Pure Math., Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 63-86.
  • [J L] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., Vol. 114, Springer-Verlag, New York, 1970.
  • [J PS S1] H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Automorphic forms on GL(3). I, II, Ann. of Math. (2) 109 (1979), 169-258.
  • [J PS S2] H. Jacquet, I. Piatetski-Shapiro and J. Shalika, On Euler products and the classification of automorphic representations. I, II, Amer. J. Math. 103 (1981), 499-558, 777-815.
  • [J PS S3] H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Relèvement cubique non normal, C. R. Acad. Sci. Paris Sér. A 292 (1981), 567-571.
  • [Kaz] D. Kazhdan, Some applications of the Weil representation, J. Analyse Math. 32 (1977), 235-248.
  • [Kn Zn] A. W. Knapp and G. Zuckerman, Normalizing factors, tempered representations, and L-groups, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 93-105.
  • [Kott] R. Kottwitz, Unstable orbital integrals on SL(3), Duke Math. J. 48 (1981).
  • [Kudla] S. Kudla, Relations between automorphic forms produced by theta-functions, Modular Functions of One Variable. VI, Lecture Notes in Math., vol. 627, Springer, New York, 1977.
  • [Lab Lang] J.-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979), 726-785.
  • [Langlands 1] R. P. Langlands, Euler products, Yale Univ. Press, New Haven, Conn., 1967.
  • [Langlands 2] R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., Vol. 170, Springer-Verlag, New York, 1970, pp. 18-86.
  • [Langlands 3] R. P. Langlands, L-functions and automorphic representations, Proc. Internat. Congress Math., Helsinki, 1978.
  • [Langlands 4] R. P. Langlands, Base change for GL2, Ann. of Math. Stud., No. 96, Princeton Univ. Press, Princeton, N. J., 1980.
  • [Langlands 5] R. P. Langlands, On the notion of an automorphic representation, Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 203-208.
  • [Langlands 6] R. P. Langlands, Les débuts d'une formule des traces stables, Lecture Notes, L'École Normale Supérieure de Jeunes Filles, Paris, 1980.
  • [Langlands 7] R. P. Langlands, Shimura varieties and the Selberg trace formula, Canad. J. Math. 29 (1977), 1292-1299.
  • [Langlands 8] R. P. Langlands, Automorphic representations, Shimura varieties and motives, Proc. Sympos. Pure Math. Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 205-246.
  • [Langlands 9] R. P. Langlands, Stable conjugacy: definitions and lemmas, Canad. J. Math. 31 (1979), 700-725.
  • [Li] W. Li, On the representations of GL(2). I, II, J. Reine Angew. Math 313 (1980), 27-42; ibid 314 (1980), 3-20; also Hecke-Weil-Jacquet-Langlands theorem revisited, Number Theory (Carbondale, 1979), Lecture Notes in Math., Vol. 751, Springer, Berlin, 1979.
  • [Lips] R. Lipsman, Group representations, Lecture Notes in Math., No. 388, Springer, New York, 1974.
  • [Mac] I. G. Macdonald, Spherical functions on a group of p-adic type, Ramanujan Inst. for Advanced Study in Math., Univ. of Madras, Madras, India, 1971.
  • [Mackey] G. Mackey, Harmonic analysis as the exploitation of symmetry–a historical survey. Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698.
  • [Mazur] B. Mazur, Review of Ernst Edward Kummer, Collected papers, Bull. Amer. Math. Soc. 83 (1977), 976-988.
  • [Mazur Wiles] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math, (submitted)
  • [Milne Shih] J. Milne and K-y. Shih, The action of complex conjugation on a Shimura variety, Ann. of Math. (2) 113 (1981), 569-599.
  • [Moy] A. Moy, Local constants and the tame Langlands correspondence, Thesis, Univ. of Chicago, 1982.
  • [Novod] M. Novodvordsky, Automorphic L-functions for the symplectic group GSp (4), Proc. Sympos. Pure Math. Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 87-95.
  • [Ogg] A. Ogg, Modular forms and Dirichlet series, Benjamin, New York, 1969.
  • [Os Wa] M. S. Osborne and G. Warner, The Selberg trace formula, I-V, Univ. of Washington, 1980-1983, preprints.
  • [PS] I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983), 309-338.
  • [PS2] I. Piatetski-Shapiro, L-functions for GSp4, preprint, 1982.
  • [PS3] I. Piatetski-Shapiro, Tate theory for reductive groups and distinguished representations, Proc. Internat. Congress Math., Helsinki, 1978.
  • [PSS] I. Piatetski-Shapiro and D. Soudry, L and e factors for GSp (4), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1982), 505-530.
  • [Plat] V. P. Platonov, The arithmetic theory of algebraic groups, Russian Math. Surveys 37 (1982), no. 3, 1-62.
  • [Ral] S. Rallis, Langlands'functorially and the Weil representation, Amer. J. Math. 104 (1982), 469-515.
  • [R S] S. Rallis and G. Schiffmann, Automorphic forms constructed from the Weil representation: Holomorphic case, Amer. J. Math. 100 (1978), 1049-1122.
  • [Rap Z] M. Rapoport and T. Zink, Uber die lokale Zeta-funktion von Shimuravarietaten, Invent. Math. 68 (1982), 21-101.
  • [Rep] J. Repka, Base change lifting and Galois invariance, Pacific J. Math. 95 (1981), 205-212; ibid. 93 (1981), 193-200.
  • [Rib] K. Ribet, A modular construction of unramified p-extensions of Q(μp), Invent. Math. 34 (1976), 151-162.
  • [Rob l] A. Robert, Formes automorphes sur GL2 [Travaux de H. Jacquet et R. P. Langlands], Semi. Bourbaki 1971/1972, no. 415, Lecture Notes in Math., Vol. 317, Springer, 1973.
  • [Rob 2] A. Robert, Des adeles: pourquoi?, L'Enseigenment Math. (2) 20 (1974), 133-146.
  • [Rob 3]A. Robert, Lectures on automorphic forms, Queens Univ., Ontario, 1976.
  • [Rog] J. Rogawski, Representations of GL(n) and division algebras over a p-adic field, Duke Math. J. 50 (1983), 161-196.
  • [Saito] H. Saito, Automorphic forms and algebraic extensions of number fields, Lectures in Math., Kyoto Univ., 1975.
  • [Sa Sh] P. Sally and J. Shalika, Characters of the discrete series of representations of SL(2) over a local field, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1231-1237; also the Plancherel formula for SL(2) over a local field, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 661-667.
  • [Selberg] A. Selberg, Harmonic analysis and discontinuous groups (= the original "trace formula "!), J. Indian Math. Soc. 20 (1956), 47-87.
  • [Serre] J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic Number Fields (A. Fröhlich, ed.), Academic Press, 1977.
  • [Sha] F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297-356.
  • [Shalika] J. Shalika, Some conjectures in class field theory, Proc. Sympos. Pure Math., Vol. 20, Amer. Math. Soc., Providence, R. I., 1971, pp. 115-122.
  • [ShaTan] J. Shalika and S. Tanaka, On an explicit construction of a certain class of automorphic forms, Amer. J. Math. 91 (1969), 1049-1076.
  • [Shel 1] D. Shelstad, Notes on L-indistinguishability (based on a lecture of R. P. Langlands), Proc. Sympos. Pure Math. Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 193-203.
  • [Shel 2] D. Shelstad, Characters and inner forms of a quasi-split group over R, Compositio Math. 39 (1979), 11-45.
  • [Shel 3] D. Shelstad, Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci. École Norm. Sup. 12 (1979), 1-31.
  • [Shel 4] D. Shelstad, Embeddings of L-groups, Canad. J. Math. 33 (1981), 513-558.
  • [Shel 5] D. Shelstad, L-indistinguishability for real groups, Math. Ann. 259 (1982), 385-430.
  • [Shimizu] H. Shimizu, Theta-series and automorphic forms on GL2, J. Math. Soc. Japan 24 (1972), 638-683.
  • [Shimura] G. Shimura, On modular forms of half-integral weight, Ann. of Math. (2) 97 (1973), 440-481.
  • [Shimura 2] G. Shimura, Correspondence modulaires et les fonctions ζ de courbes algébriques, J. Math. Soc. Japan 10 (1958), 1-28.
  • [Shimura 3] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan, No. 11, Princeton Univ. Press, Princeton, N. J., 1971.
  • [Shintani] T. Shintani, On liftings of holomorphic cusp forms, Proc. Sympos. Pure Math., Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 97-110.
  • [Sil] A. Silberger, Discrete series and classification for p-adic groups. I, Amer. J. Math. 103 (1981), 1241-1321.
  • [Springer] T. A. Springer, Reductive groups Proc. Sympos. Pure Math., Vol. 33, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 3-28.
  • [Tate] J. Tate, Problem 9: the general reciprocity law, Proc. Sympos. Pure Math., Vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 311-322.
  • [Tate 2] J. Tate, Number theoretic background, Proc. Symos. Pure Math., Vol. 33, part 2, Amer. Math. Soc., Providence, R. I., 1979, pp. 3-26.
  • [Ti] J. Tits, Reductive groups over local fields Proc. Sympos. Pure Math., Vol. 33, part I, Amer. Math. Soc., Providence, R. I., 1979, pp. 29-70.
  • [Tun] J. Tunnell, Artin's conjecture for representations or octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 173-175.
  • [Vigneras] M.-F. Vigneras, Charactérisation des integrales orbitales sur un groupe réductife p-adique, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 945-961 (1982).
  • [Wald] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. 59 (1980), 1-133.
  • [Warner] G. Warner, Harmonic analysis on semi-simple Lie groups. I, II, Springer-Verlag, Berlin, 1972.
  • [We 1] A. Weil, Über die bestimmung Dirichletscher Rieben durch funktionelgleichungen, Math. Ann. 168 (1967), 149-156; also Dirichlet series and automorphic forms, Lecture Notes in Math., Vol. 189, Springer-Verlag, New York, 1971.
  • [We 2] A. Weil, Sur certaines groups d'operateurs unitaires, Acta Math. 111 (1964), 143-211.
  • [We 3] A. Weil, Basic number theory, 3rd ed., Springer-Verlag, New York, 1974.
  • [Zagier] D. Zagier, Eisenstein series and the Selberg trace formula. I, Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Springer-Verlag, Berlin, 1981, pp. 275-301.