Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

   
 
 

 

An elementary introduction to the Langlands program


Author: Stephen Gelbart
Journal: Bull. Amer. Math. Soc. 10 (1984), 177-219
MSC (1980): Primary 10D40, 12A67; Secondary 22E55
DOI: https://doi.org/10.1090/S0273-0979-1984-15237-6
Link to commentary: Bull. Amer. Math. Soc., 48 (2011), no. 4, 513--535.
MathSciNet review: 733692
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • A. N. Andrianov, Zeta functions and the Siegel modular forms, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 9–20. MR 0404149
  • James Arthur, Automorphic representations and number theory, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 3–51. MR 670091
  • James Arthur, Eisenstein series and the trace formula, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 253–274. MR 546601
  • [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.
  • James Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), no. 1, 1–74. MR 625344, DOI https://doi.org/10.2307/1971376
  • Tetsuya Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), no. 1, 81–94. MR 429751, DOI https://doi.org/10.1007/BF01391220
  • I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 0425030
  • H. P. F. Swinnerton-Dyer and B. J. Birch, Elliptic curves and modular functions, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 2–32. Lecture Notes in Math., Vol. 476. MR 0384813
  • A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR 546608
  • Armand Borel, Formes automorphes et séries de Dirichlet (d’après R. P. Langlands), Séminaire Bourbaki (1974/1975: Exposés Nos. 453–470), Exp. No. 466, Springer, Berlin, 1976, pp. 183–222. Lecture Notes in Math., Vol. 514 (French). MR 0447118
  • A. Borel and H. Jacquet, Automorphic forms and automorphic representations, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 189–207. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. MR 546598
  • A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
  • Pierre Cartier, La conjecture locale de Langlands pour ${\rm GL}(2)$ et la démonstration de Ph. Kutzko, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 112–138 (French). MR 636520
  • P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
  • W. Casselman, The Hasse-Weil $\zeta $-function of some moduli varieties of dimension greater than one, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 141–163. MR 546615
  • W. Casselman, ${\rm GL}_{n}$, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 663–704. MR 0562502
  • J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
  • [Cas Fro] J. W. Cassels and A. Frölich (Editors), Algebraic number theory, Proc. Instructional Conf. (Brighton), Thompson Book, Washington, D. C, 1967.
  • Laurent Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 1, 45–115 (French). MR 672475
  • John Coates, The work of Mazur and Wiles on cyclotomic fields, Bourbaki Seminar, Vol. 1980/81, Lecture Notes in Math., vol. 901, Springer, Berlin-New York, 1981, pp. 220–242. MR 647499
  • P. Deligne, Formes modulaires et représentations de ${\rm GL}(2)$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 55–105. Lecture Notes in Math., Vol. 349 (French). MR 0347738
  • P. Deligne, Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 501–597. Lecture Notes in Math., Vol. 349 (French). MR 0349635
  • Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379
  • [Drin] V. G. Drinfeld, Langlands conjecture for GL(2) over function fields, Proc. Internat. Congress. Math., Helsinki, 1978.
  • B. Dwork, On the Artin root number, Amer. J. Math. 78 (1956), 444–472. MR 82476, DOI https://doi.org/10.2307/2372524
  • Daniel Flath, A comparison of the automorphic representations of ${\rm GL}(3)$ and its twisted forms, Pacific J. Math. 97 (1981), no. 2, 373–402. MR 641166
  • [Flick 1] Y. Flicker, The adjoint lifting from SL(2) to PGL(3), I.H.E.S., 1981, preprint.
  • Yuval Z. Flicker, Packets and liftings for ${\rm U}(3)$, J. Analyse Math. 50 (1988), 19–63. MR 942819, DOI https://doi.org/10.1007/BF02796113
  • Yuval Z. Flicker, Automorphic forms on covering groups of ${\rm GL}(2)$, Invent. Math. 57 (1980), no. 2, 119–182. MR 567194, DOI https://doi.org/10.1007/BF01390092
  • Yuval Z. Flicker, The trace formula and base change for ${\rm GL}(3)$, Lecture Notes in Mathematics, vol. 927, Springer-Verlag, Berlin-New York, 1982. MR 663002
  • Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. Annals of Mathematics Studies, No. 83. MR 0379375
  • Stephen Gelbart, Elliptic curves and automorphic representations, Advances in Math. 21 (1976), no. 3, 235–292. MR 439754, DOI https://doi.org/10.1016/S0001-8708%2876%2980001-1
  • Stephen Gelbart, Automorphic forms and Artin’s conjecture, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn., Bonn, 1976) Springer, Berlin, 1977, pp. 241–276. Lecture Notes in Math., Vol. 627. MR 0568306
  • Stephen Gelbart, Examples of dual reductive pairs, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 287–296. MR 546603
  • Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066
  • Stephen Gelbart and Hervé Jacquet, Forms of ${\rm GL}(2)$ from the analytic point of view, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 213–251. MR 546600
  • S. S. Gelbart and A. W. Knapp, $L$-indistinguishability and $R$ groups for the special linear group, Adv. in Math. 43 (1982), no. 2, 101–121. MR 644669, DOI https://doi.org/10.1016/0001-8708%2882%2990030-5
  • 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. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 1–39. MR 633657
  • Stephen Gelbart and Ilya Piatetski-Shapiro, Automorphic forms and $L$-functions for the unitary group, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 141–184. MR 748507, DOI https://doi.org/10.1007/BFb0073147
  • I. M. Gel′fand and S. V. Fomin, Geodesic flows on manifolds of constant negative curvature, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 1(47), 118–137 (Russian). MR 0052701
  • I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR 0233772
  • I. M. Gel′fand and D. A. Kajdan, Representations of the group ${\rm GL}(n,K)$ where $K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 95–118. MR 0404534
  • Paul Gérardin, Cuspidal unramified series for central simple algebras over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 157–169. MR 546594
  • P. Gérardin and J.-P. Labesse, The solution of a base change problem for ${\rm GL}(2)$ (following Langlands, Saito, Shintani), Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 115–133. MR 546613
  • Roger Godement, La formule des traces de Selberg considérée comme source de problèmes mathématiques [ MR0182681 (32 #164)], Séminaire Bourbaki, Vol. 8, Soc. Math. France, Paris, 1995, pp. Exp. No. 244, 53–62 (French). MR 1611530
  • [God 2] R. Godement, Introduction aux travaux de Selberg, Sem. Bourbaki, 9e année: 1956/1957, No. 144.
  • Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
  • Larry Joel Goldstein, Analytic number theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0498335
  • Kenneth I. Gross, On the evolution of noncommutative harmonic analysis, Amer. Math. Monthly 85 (1978), no. 7, 525–548. MR 505524, DOI https://doi.org/10.2307/2320861
  • Benedict H. Gross, On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication, Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math., vol. 26, Birkhäuser, Boston, Mass., 1982, pp. 219–236. MR 685298
  • G. Harder, Chevalley groups over function fields and automorphic forms, Ann. of Math. (2) 100 (1974), 249–306. MR 563090, DOI https://doi.org/10.2307/1971073
  • [HC] Harish-Chandra, Automorphic forms on semi-simple Lie groups, Lecture Notes in Math., Vol. 62, Springer, New York, 1968.
  • Guy Henniart, La conjecture de Langlands locale pour ${\rm GL}(3)$, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981) Progr. Math., vol. 22, Birkhäuser, Boston, Mass., 1982, pp. 91–105 (French). MR 693313
  • R. Howe, $\theta $-series and invariant theory, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285. MR 546602
  • Roger E. Howe, Tamely ramified supercuspidal representations of ${\rm Gl}_{n}$, Pacific J. Math. 73 (1977), no. 2, 437–460. MR 492087
  • Roger E. Howe, Some qualitative results on the representation theory of ${\rm Gl}_{n}$ over a $p$-adic field, Pacific J. Math. 73 (1977), no. 2, 479–538. MR 492088
  • James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
  • Hervé Jacquet, Représentations des groupes linéaires $p$-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 (French). MR 0291360
  • Hervé Jacquet, Automorphic forms on ${\rm GL}(2)$. Part II, Lecture Notes in Mathematics, Vol. 278, Springer-Verlag, Berlin-New York, 1972. MR 0562503
  • Hervé Jacquet, Principal $L$-functions of the linear group, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 63–86. MR 546609
  • H. Jacquet and R. P. Langlands, Automorphic forms on ${\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
  • Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika, Automorphic forms on ${\rm GL}(3)$. I, Ann. of Math. (2) 109 (1979), no. 1, 169–212. MR 519356, DOI https://doi.org/10.2307/1971270
  • [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.
  • Hervé Jacquet, Ilja I. Piatetski-Shapiro, and Joseph Shalika, Relèvement cubique non normal, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 12, 567–571 (French, with English summary). MR 615450
  • David Kazhdan, Some applications of the Weil representation, J. Analyse Mat. 32 (1977), 235–248. MR 0492089, DOI https://doi.org/10.1007/bf02803582
  • A. W. Knapp and Gregg Zuckerman, Normalizing factors, tempered representations, and $L$-groups, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 93–105. MR 546591
  • Robert E. Kottwitz, Unstable orbital integrals on ${\rm SL}(3)$, Duke Math. J. 48 (1981), no. 3, 649–664. MR 630589
  • Stephen S. Kudla, Relations between automorphic forms produced by theta-functions, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 277–285. Lecture Notes in Math., Vol. 627. MR 0480343
  • J.-P. Labesse and R. P. Langlands, $L$-indistinguishability for ${\rm SL}(2)$, Canadian J. Math. 31 (1979), no. 4, 726–785. MR 540902, DOI https://doi.org/10.4153/CJM-1979-070-3
  • Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967; Yale Mathematical Monographs, 1. MR 0419366
  • R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 18–61. Lecture Notes in Math., Vol. 170. MR 0302614
  • [Langlands 3] R. P. Langlands, L-functions and automorphic representations, Proc. Internat. Congress Math., Helsinki, 1978.
  • Robert P. Langlands, Base change for ${\rm GL}(2)$, Annals of Mathematics Studies, No. 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 574808
  • R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
  • [Langlands 6] R. P. Langlands, Les débuts d’une formule des traces stables, Lecture Notes, L’École Normale Supérieure de Jeunes Filles, Paris, 1980.
  • R. P. Langlands, Shimura varieties and the Selberg trace formula, Canadian J. Math. 29 (1977), no. 6, 1292–1299. MR 498400, DOI https://doi.org/10.4153/CJM-1977-129-2
  • R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
  • R. P. Langlands, Stable conjugacy: definitions and lemmas, Canadian J. Math. 31 (1979), no. 4, 700–725. MR 540901, DOI https://doi.org/10.4153/CJM-1979-069-2
  • [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.
  • Ronald L. Lipsman, Group representations, Lecture Notes in Mathematics, Vol. 388, Springer-Verlag, Berlin-New York, 1974. A survey of some current topics. MR 0372116
  • I. G. Macdonald, Spherical functions on a group of $p$-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the Ramanujan Institute, No. 2. MR 0435301
  • George W. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 1, 543–698. MR 571370, DOI https://doi.org/10.1090/S0273-0979-1980-14783-7
  • [Mazur] B. Mazur, Review of Ernst Edward Kummer, Collected papers, Bull. Amer. Math. Soc. 83 (1977), 976-988.
  • B. Mazur and A. Wiles, Class fields of abelian extensions of ${\bf Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI https://doi.org/10.1007/BF01388599
  • J. S. Milne and Kuang-yen Shih, The action of complex conjugation on a Shimura variety, Ann. of Math. (2) 113 (1981), no. 3, 569–599. MR 621017, DOI https://doi.org/10.2307/2006998
  • [Moy] A. Moy, Local constants and the tame Langlands correspondence, Thesis, Univ. of Chicago, 1982.
  • Mark E. Novodvorsky, Automorphic $L$-functions for symplectic group ${\rm GSp}(4)$, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 87–95. MR 546610
  • Andrew Ogg, Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0256993
  • [Os Wa] M. S. Osborne and G. Warner, The Selberg trace formula, I-V, Univ. of Washington, 1980-1983, preprints.
  • I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting, Invent. Math. 71 (1983), no. 2, 309–338. MR 689647, DOI https://doi.org/10.1007/BF01389101
  • [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.
  • Ilya I. Piatetski-Shapiro and David Soudry, $L$ and $\varepsilon $ factors for ${\rm GSp}(4)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 505–530 (1982). MR 656034
  • V. P. Platonov, Arithmetic theory of algebraic groups, Uspekhi Mat. Nauk 37 (1982), no. 3(225), 3–54, 224 (Russian). MR 659426
  • Stephen Rallis, Langlands’ functoriality and the Weil representation, Amer. J. Math. 104 (1982), no. 3, 469–515. MR 658543, DOI https://doi.org/10.2307/2374151
  • S. Rallis and G. Schiffmann, Automorphic forms constructed from the Weil representation: holomorphic case, Amer. J. Math. 100 (1978), no. 5, 1049–1122. MR 517145, DOI https://doi.org/10.2307/2373962
  • M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), no. 1, 21–101 (German). MR 666636, DOI https://doi.org/10.1007/BF01394268
  • Joe Repka, Base change lifting and Galois invariance, Pacific J. Math. 95 (1981), no. 1, 205–212. MR 631670
  • Kenneth A. Ribet, A modular construction of unramified $p$-extensions of $Q(\mu _{p})$, Invent. Math. 34 (1976), no. 3, 151–162. MR 419403, DOI https://doi.org/10.1007/BF01403065
  • Alain Robert, Formes automorphes sur ${\rm GL}_{2}$, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 415, Springer, Berlin, 1973, pp. 295–318. Lecture Notes in Math., Vol. 317 (French). MR 0424697
  • Alain Robert, Des adèles: pourquoi?, Enseign. Math. (2) 20 (1974), 133–145 (French). MR 357369
  • [Rob 3]A. Robert, Lectures on automorphic forms, Queens Univ., Ontario, 1976.
  • Jonathan D. Rogawski, Representations of ${\rm GL}(n)$ and division algebras over a $p$-adic field, Duke Math. J. 50 (1983), no. 1, 161–196. MR 700135
  • Hiroshi Saito, Automorphic forms and algebraic extensions of number fields, Proc. Japan Acad. 51 (1975), no. 4, 229–233. MR 384703
  • P. J. Sally Jr. and J. A. Shalika, Characters of the discrete series of representations of ${\rm SL}(2)$ over a local field, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1231–1237. MR 237713, DOI https://doi.org/10.1073/pnas.61.4.1231
  • A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
  • J.-P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 193–268. MR 0450201
  • Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI https://doi.org/10.2307/2374219
  • J. A. Shalika, Some conjectures in class field theory, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 115–122. MR 0345935
  • J. A. Shalika and S. Tanaka, On an explicit construction of a certain class of automorphic forms, Amer. J. Math. 91 (1969), 1049–1076. MR 291087, DOI https://doi.org/10.2307/2373316
  • D. Shelstad, Notes on $L$-indistinguishability (based on a lecture of R. P. Langlands), Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 193–203. MR 546618
  • D. Shelstad, Characters and inner forms of a quasi-split group over ${\bf R}$, Compositio Math. 39 (1979), no. 1, 11–45. MR 539000
  • Diana Shelstad, Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 1–31. MR 532374
  • D. Shelstad, Embeddings of $L$-groups, Canadian J. Math. 33 (1981), no. 3, 513–558. MR 627641, DOI https://doi.org/10.4153/CJM-1981-044-4
  • D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385–430. MR 661206, DOI https://doi.org/10.1007/BF01456950
  • Hideo Shimizu, Theta series and automorphic forms on ${\rm GL}_{2}$, J. Math. Soc. Japan 24 (1972), 638–683. MR 333081, DOI https://doi.org/10.2969/jmsj/02440638
  • Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI https://doi.org/10.2307/1970831
  • Goro Shimura, Correspondances modulaires et les fonctions $\zeta $ de courbes algébriques, J. Math. Soc. Japan 10 (1958), 1–28 (French). MR 95173, DOI https://doi.org/10.2969/jmsj/01010001
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR 0314766
  • Takuro Shintani, On liftings of holomorphic cusp forms, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 97–110. MR 546611
  • Allan J. Silberger, Discrete series and classification for $p$-adic groups. I, Amer. J. Math. 103 (1981), no. 6, 1241–1321. MR 636960, DOI https://doi.org/10.2307/2374232
  • T. A. Springer, Reductive groups, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR 546587
  • J. Tate, Problem 9: The general reciprocity law, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 311–322. Proc. Sympos. Pure Math., Vol. XXVIII. MR 0429839
  • J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
  • J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
  • Jerrold Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173–175. MR 621884, DOI https://doi.org/10.1090/S0273-0979-1981-14936-3
  • Marie-France Vignéras, Caractérisation des intégrales orbitales sur un groupe réductif $p$-adique, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 945–961 (1982) (French). MR 656066
  • J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1–132 (French). MR 577010
  • Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. MR 0498999
  • André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156 (German). MR 207658, DOI https://doi.org/10.1007/BF01361551
  • André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI https://doi.org/10.1007/BF02391012
  • André Weil, Basic number theory, 3rd ed., Springer-Verlag, New York-Berlin, 1974. Die Grundlehren der Mathematischen Wissenschaften, Band 144. MR 0427267
  • D. Zagier, Eisenstein series and the Riemann zeta function, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 275–301. MR 633666

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 10D40, 12A67, 22E55

Retrieve articles in all journals with MSC (1980): 10D40, 12A67, 22E55


Additional Information