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A Note on the Automorphic Langlands Group

Published online by Cambridge University Press:  20 November 2018

James Arthur
James Arthur*
Affiliation:
Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3
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Abstract

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Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

Keywords

11R3922E55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 4 , 01 December 2002 , pp. 466 - 482
Copyright
Copyright © Canadian Mathematical Society 2002

References

[A] Arthur, J., The principle of functoriality. Bull. Amer. Math. Soc., to appear.Google Scholar
[Bl] Blasius, D., On multiplicities for SL(n). Israel J. Math. 88 (1994), 237–251.Google Scholar
[Bo] Borel, A., Automorphic L-functions. In: Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. (2) 33, Amer.Math. Soc., 1979, 27–61.Google Scholar
[C] Clozel, L., Motifs et formes automorphes: Applications du principe de fonctorialité. In: Automorphic Forms, Shimura Varieties, and L-functions, Perspect.Math. (1) 11, Academic Press, 1990, 77–159.Google Scholar
[D] Deligne, P., Motifs et groupe de Taniyama. Lecture Notes in Math. 900, Springer-Verlag, New York, 1982, 261–297.Google Scholar
[DM] Deligne, P. and Milne, J., Tannakian categories. Lecture Notes in Math. 900, Springer-Verlag, New York, 1982, 101–228.Google Scholar
[HT] Harris, M. and Taylor, R., On the geometry and cohomology of some simple Shimura varieties. Ann. of Math. Studies 151, Princeton University Press, Princeton, 2001.Google Scholar
[H] Henniart, G., Une preuve simple des conjectives de Langlands de GL(n) sur un corps p-adique. Invent.Math. 139 (2000), 439–455.Google Scholar
[J] Jacquet, H., Principal L-functions of the linear group. In: Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. (2) 33, Amer. Math. Soc. 1979, 63–86.Google Scholar
[Klei] Kleiman, S., The standard conjectures. In: Motives, Proc. Sympos. Pure Math. (1) 55, Amer. Math. Soc. 1994, 3–20.Google Scholar
[K1] Kottwitz, R., Rational conjugacy classes in reductive groups. Duke Math. J. 49 (1982), 785–806.Google Scholar
[K2] Kottwitz, R., Stable trace formula: cuspidal tempered terms. Duke Math. J. 51 (1984), 611–650.Google Scholar
[KS] Kottwitz, R. and Shelstad, D., Foundations of Twisted Endoscopy. Ast érisque 255, 1999, Soc. Math. de France.Google Scholar
[Lab] Labesse, J.-P., Cohomologie, L-groupes et functorialité. Compositio Math. 55 (1985), 163–184.Google Scholar
[LL] Labesse, J.-P. and Langlands, R., L-indistinguishability for SL2. Canad. J. Math. 31 (1979), 726–785.Google Scholar
[L1] Langlands, R., Problems in the theory of automorphic forms. Lecture Notes in Math. 170, Springer-Verlag, 1970, 18–86.Google Scholar
[L2] Langlands, R., Representations of abelian algebraic groups. Pacific J. Math. (1997), Special Issue, 231–250.Google Scholar
[L3] Langlands, R., On the classification of irreducible representations of real algebraic groups. In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Amer.Math. Soc. Math. Surveys and Monographs 31, 1989, 101–170.Google Scholar
[L4] Langlands, R., On the notion of an automorphic representation. In: Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math. (1) 33, Amer.Math. Soc. 1979, 203–208.Google Scholar
[L5] Langlands, R., Automorphic representations, Shimura varieties, and motives. Ein Märchen. In: Automorphic Forms, Representations and L-functions, Pure Math. (2) 33, Amer.Math. Soc. 1979, 205–246.Google Scholar
[L6] Langlands, R., Stable conjugacy: definitions and lemmas. Canad. J. Math. 31 (1979), 700–725.Google Scholar
[L7] Langlands, R., Beyond endoscopy. Institute for Advanced Study, Princeton, (2001), preprint.Google Scholar
[LS] Langlands, R. and Shelstad, D., On the definition of transfer factors.Math. Ann. 278 (1987), 219–271.Google Scholar
[Lars1] Larsen, M., On the conjugacy of element-conjugate homomorphisms. Israel J. Math. 88 (1994), 253–277.Google Scholar
[Lars2] Larsen, M., On the conjugacy of element conjugate homomorphisms II. Quart. J. Math. Oxford Ser. (2) 47 (1996), 73–85.Google Scholar
[MS] Milne, J. and Shih, K.-Y., Langlands’ construction of the Taniyama group. Lecture Notes in Math 900, Springer Verlag, New York, 1982, 229–260.Google Scholar
[R] Ramakrishnan, D., Pure motives and automorphic forms. In: Motives, Proc. Sympos. Pure Math. (2) 55, Amer.Math. Soc. 1994, 411–446.Google Scholar
[S1] Serre, J.-P., Abelian l-adic Representations and Elliptic Curves. Benjamin, New York, 1968.Google Scholar
[S2] Serre, J.-P., Propriétés conjecturales des groups de Galois motiviques et representations l-adiques. In: Motives, Proc. Sympos. Pure Math. (1) 55, Amer.Math. Soc. 1994, 377–400.Google Scholar
[T] Tate, J., Number theoretic background. In: Automorphic Forms, Representations and L-functions, Proc. Sympos. Pure. Math. (2) 33, Amer. Math. Soc. 1979, 3–26.Google Scholar
[W] Wang, S., in preparation.Google Scholar