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The Trace Formula and Its Applications: An Introduction to the Work of James Arthur

Published online by Cambridge University Press:  20 November 2018

Robert P. Langlands
Robert P. Langlands*
Affiliation:
School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ 08540 USA
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In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a high honour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to a mathematician, and the choice of Arthur, although certainly a recognition of his greatmerits, is also a recognition of the vigour of contemporary Canadian mathematics.

Keywords

11F7011F7258G25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 2 , 01 June 2001 , pp. 160 - 209
Copyright
Copyright © Canadian Mathematical Society 2001

References

Bibliography of James G. Arthur

[1] The Selberg trace formula for groups of F-rank one. Ann. of Math. 100 (1974), 326–385.Google Scholar
[2] Some tempered distributions on groups of real rank one. Ann. of Math. 100 (1974), 553–584.Google Scholar
[3] A theorem on the Schwartz space of a reductive Lie group. Proc. Nat. Acad. Sci. 72 (1975), 4718–4719.Google Scholar
[4] The characters of discrete series as orbital integrals. Invent.Math. 32 (1976), 205–261.Google Scholar
[5] A truncation process for reductive groups. Bull. Amer. Math. Soc. 83 (1977), 748–750.Google Scholar
[6] Eisenstein series and the trace formula. Proc. Sympos. Pure Math. 33(1979), Part 1, 253–274.Google Scholar
[7] A trace formula for reductive groups I: Terms associated to classes in G(Q). Duke Math. J. 245 (1978), 911–952.Google Scholar
[8] Harmonic analysis of invariant distributions. Lie theories and their applications (Queen's University, 1977), Queen's Papers in Pure and Appl. Math. 48 (1978), 384–393.Google Scholar
[9] A trace formula for reductive groups II: Applications of a truncation operator. Compositio Math. 40 (1980), 87–121.Google Scholar
[10] The trace formula in invariant form. Ann. of Math. 114 (1981), 1–74.Google Scholar
[11] Automorphic representations and number theory. C.M.S. Conf. Proc. 1 (1981), 3–51.Google Scholar
[12] On the inner product of truncated Eisenstein series. Duke Math. J. 49 (1982), 35–70.Google Scholar
[13] On a family of distributions obtained from Eisenstein series I: Application of the Paley-Wiener theorem. Amer. J. Math. 104 (1982), 1243–1288.Google Scholar
[14] On a family of distributions obtained from Eisenstein series II: Explicit formulas. Amer. J. Math. 104 (1982), 1289–1336.Google Scholar
[15] A Paley-Wiener theorem for real reductive groups. Acta Math. 150 (1983), 1–89.Google Scholar
[16] The trace formula for reductive groups. Publ. Math. Univ. Paris VII 15 (1983), 1–41.Google Scholar
[17] Multipliers and a Paley-Wiener theorem for real reductive groups. Representation Theory of Reductive Groups (Park City, Utah, 1982), Progr. Math. 40 (1983), 1–19.Google Scholar
[18] On some problems suggested by the trace formula. Lie Group Representations II, Springer Lecture Notes in Math. 1041 (1984), 1–49.Google Scholar
[19] The trace formula for noncompact quotient. In: Proceedings of International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw, 1984, 849–859.Google Scholar
[20] A measure on the unipotent variety. Canad. J. Math. 37 (1985), 1237–1274.Google Scholar
[21] On a family of distributions obtained from orbits. Canad. J. Math. 38 (1986), 179–214.Google Scholar
[22] The Fourier transform of weighted orbital integrals on SL(2, R).With R. Herb and P. Sally. Contemp. Math. 53 (1986), 17–37.Google Scholar
[23] The characters of supercuspidal representations as weighted orbital integrals. Proc. Indian Acad. Sci. 97 (1987), 3–19.Google Scholar
[24] The local behaviour of weighted orbital integrals. Duke Math. J. 56 (1988), 223–293.Google Scholar
[25] Characters, harmonic analysis, and an L2-Lefschetz formula. Proc. Sympos. Pure Math. 48 (1988), 167–179.Google Scholar
[26] The invariant trace formula I. Local theory. J. Amer. Math. Soc. 1 (1988), 323–383.Google Scholar
[27] The invariant trace formula II. Global theory. J. Amer. Math. Soc. 1 (1988), 501–554.Google Scholar
[28] Intertwining operators and residues I.Weighted characters. J. Funct. Anal. 84 (1989), 19–84.Google Scholar
[29] Intertwining operators and residues II. Invariant distributions. Compositio Math. 70 (1989), 51–99.Google Scholar
[30] Simple Algebras, Base Change and the Advanced Theory of the Trace Formula.With L. Clozel. Ann. of Math. Stud. 120, Princeton University Press, 1989.Google Scholar
[31] The trace formula and Hecke operators. In: Number Theory, Trace Formulas and Discrete Groups, Academic Press, 1989, 11–27.Google Scholar
[32] The L2-Lefschetz numbers of Hecke operators. Invent.Math. 97 (1989), 257–290.Google Scholar
[33] Harmonic analysis of tempered distributions on semisimple Lie groups of real rank one. In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups (eds. P. Sally and D. Vogan),Math. SurveysMonographs 31, Amer. Math. Soc., 1989, 13–100.Google Scholar
[34] Unipotent automorphic representations: Conjectures. Astérisque 171–172 (1989), 13–71.Google Scholar
[35] Unipotent automorphic representations: Global Motivation. In: Automorphic Forms, Shimura Varieties and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect.Math. 10 (1989), 1–75.Google Scholar
[36] Towards a local trace formula. In: Proceedings of First JAMI Conference, Johns Hopkins University Press, 1990, 1–23.Google Scholar
[37] Lectures on automorphic L-functions. With S. Gelbart. L-functions and Arithmetic (Durham, 1990), London Math. Soc. Lecture Note Ser. 153 (1991), 1–59.Google Scholar
[38] Some problems in local harmonic analysis. Harmonic Analysis on Reductive Groups (Brunswick, ME, 1989), Progr.Math. 101 (1991), 57–78.Google Scholar
[39] A local trace formula. Inst. Hautes Études Sci. Publ. Math. 73 (1991), 5–96.Google Scholar
[40] On elliptic tempered characters. Acta Math. 171 (1993), 73–138.Google Scholar
[41] On the Fourier transforms of weighted orbital integrals. J. Reine Angew. Math. 452 (1994), 163–217.Google Scholar
[42] The trace Paley-Wiener theorem for Schwartz functions. Contemp.Math. 177 (1994), 171–180.Google Scholar
[43] L2-cohomology and automorphic representations. In: Canadian Mathematical Society 1945–1995, Vol. 3, Canadian Math. Soc., Ottawa, 1996, 1–17.Google Scholar
[44] On local character relations. Selecta Math. 2 (1996), 501–579.Google Scholar
[45] The problem of classifying automorphic representations of classical groups. CRM Proc. Lecture Notes 11 (1997), 1–12.Google Scholar
[46] Stability and endoscopy: informal motivation. Proc. Sympos. Pure Math. 61 (1997), 433–442.Google Scholar
[47] Canonical normalization of weighted characters and a transfer conjecture. C. R. Math. Rep. Acad. Sci. Canada 20 (1998), 35–52.Google Scholar
[48] Towards a stable trace formula. In: Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, 507–517.Google Scholar
[49] On the transfer of distributions: weighted orbital integrals. Duke Math. J. 99 (1999), 209–283.Google Scholar
[50] Endoscopic L-functions and a combinatorial identity. Canad. J. Math. 51 (1999), 1135–1148.Google Scholar
[51] Stabilization of a family of differential equations. Mathematical Legacy of Harish-Chandra (Baltimore, MD, 1998), Proc. Sympos. Pure Math. 68 (2000), 77–95.Google Scholar
[52] Harmonic analysis and group representations. Notices Amer. Math. Soc. 47 (2000), 26–34.Google Scholar
[53] A stable trace formula I. General expansions. Preprint.Google Scholar
[54] A stable trace formula II. Global descent. Invent.Math., to appear.Google Scholar
[55] A stable trace formula III. Proof of the main theorems. Preprint.Google Scholar
[56] The principle of functoriality. Submitted to Mathematical Challenges of the 21st Century.Google Scholar

Supplementary References

[ABV] Adams, J., Barbasch, D. and Vogan, D., The Langlands classification and irreducible characters for real reductive groups. Progr. Math. 104, Birkhäuser, 1992.Google Scholar
[CD] Clozel, L. and Delorme, P., Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. Invent.Math. 77 (1984), 427–453; Sur le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs réels. C. R. Acad. Sci. Paris. Sér. 1 Math. 300 (1985), 331–333.Google Scholar
[HPS] Howe, R. and Piatetski-Shapiro, I., A counterexample to the “generalized Ramanujan conjecture” for (quasi-)split groups. Automorphic forms, representations and L-functions (Corvallis, OR, 1977), Part 1, Proc. Sympos. Pure Math. 33 (1979), 315–322.Google Scholar
[J] Jacquet, H., On the residual spectrum of GL(n). Lie Group Representations II, Springer Lecture Notes in Math. 1041 (1984), 185–208.Google Scholar
[JL] Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2). Springer Lecture Notes in Math. 114, 1970.Google Scholar
[K1] Kottwitz, R. E., Tamagawa numbers. Ann. of Math. 127 (1988), 629–646.Google Scholar
[K2] Kottwitz, R. E., On the ƛ-adic representations associated to some simple Shimura varieties. Invent.Math. 108 (1992), 653–665 Google Scholar
[KS] Kottwitz, R. and Shelstad, D., Foundations of twisted endoscopy. Astérisque 255, 1999.Google Scholar
[LL] Labesse, J.-P. and Langlands, R. P., L-indistinguishability for SL(2). Canad. J. Math. 31 (1979), 726–785.Google Scholar
[KL] Langlands, R., Les débuts d’une formule des traces stables. Publ. Math. Univ. Paris VII 13, Paris, 1983; R. Kottwitz, Stable trace formula: cuspidal tempered terms. Duke Math. J. 51 (1984), 611–650; , Stable trace formula: elliptic singular terms. Math. Ann. 275 (1986), 365–399.Google Scholar
[L] Langlands, R. P., Rank-one residues of Eisenstein series. Israel Math, Conf. Proc. 3 (1990), 111–125.Google Scholar
[LS] Langlands, R. P. and Shelstad, D., On the definition of transfer factors.Math. Ann. 278 (1987), 219–271.Google Scholar
[Mo] Moeglin, C., Représentations unipotentes et formes automorphes de carré intégrable. Forum Math. 6 (1994), 651–744.Google Scholar
[MW] Moeglin, C. and Waldspurger, J.-L., Le spectre résiduel de GL(n). Ann. Sci. École Norm. Sup. 22 (1989), 605–674.Google Scholar
[Mü] Müller, W., The trace class conjecture in the theory of automorphic forms. Ann. of Math. 130 (1989), 473–529.Google Scholar
[R] Rogawski, J., Automorphic representations of unitary groups in three variables. Ann. of Math. Stud. 123, Princeton University Press, 1990; The multiplicity formula for A-packets. In: The zeta functions of Picard modular surfaces (eds. R. P. Langlands and D. Ramakrishnan), Centre de recherches mathématiques, Univ. de Montréal, 1992, 395–419.Google Scholar
[S] Saito, H., Automorphic forms and extensions of number fields. Lectures in Mathematics 8, Kyoto University, 1975.Google Scholar
[W] Waldspurger, J.-L., Le lemme fondamental implique le transfert. Compositio Math. 105 (1997), 153–236.Google Scholar