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Selberg’s conjectures and Artin \(L\)-functions. (English) Zbl 0805.11062

Selberg’s conjectures describe Dirichlet series with an analytic continuation, functional equation, Euler product and a Ramanujan hypothesis. The conjectures and some of their consequences are described in J. B. Conrey and A. Ghosh [Duke Math. J. 72, 673-693 (1993; Zbl 0796.11037)]. Roughly speaking the conjectures can be viewed as an alternative to the Langlands programme, but with a more analytic flavour. The present paper strengthens this connection by showing that the Selberg conjectures imply Artin’s conjecture on the holomorphy of \(L\)-functions attached to nontrivial irreducible representations of finite Galois extensions. Indeed it is shown that the Langlands reciprocity conjecture also follows, for those extensions \(K/k\) with \(K/ \mathbb{Q}\) solvable.

MSC:

11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
11R39 Langlands-Weil conjectures, nonabelian class field theory

Citations:

Zbl 0796.11037
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References:

[1] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.10022
[2] Emil Artin, The collected papers of Emil Artin, Edited by Serge Lang and John T. Tate, Addison – Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. · Zbl 0146.00101
[3] S. Bochner, On Riemann’s functional equation with multiple Gamma factors, Ann. of Math. (2) 67 (1958), 29 – 41. · Zbl 0082.29002 · doi:10.2307/1969923
[4] Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. · Zbl 0443.22010
[5] J. B. Conrey and A. Ghosh, On the Selberg class of Dirichlet series: small degrees, Duke Math. J. 72 (1993), no. 3, 673 – 693. · Zbl 0796.11037 · doi:10.1215/S0012-7094-93-07225-0
[6] -, Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., vol. 33, Amer. Math. Soc., Providence, RI, 1979.
[7] D. Flath, Decomposition of representations into tensor products, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 179 – 183. · Zbl 0414.22019
[8] Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. · Zbl 0244.12011
[9] J. Hoffstein and R. Murty, L-series of automorphic forms on \( GL{_3}(R)\), Number Theory , Walter de Gruyter, Berlin and New York, 1989. · Zbl 0686.10021
[10] H. Jacquet and R. P. Langlands, Automorphic forms on \?\?(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. · Zbl 0236.12010
[11] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499 – 558. , https://doi.org/10.2307/2374103 H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777 – 815. · Zbl 0491.10020 · doi:10.2307/2374050
[12] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499 – 558. , https://doi.org/10.2307/2374103 H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777 – 815. · Zbl 0491.10020 · doi:10.2307/2374050
[13] 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. · Zbl 0225.14022
[14] -, Base change for \( GL(2)\), Ann. of Math. Stud., vol. 96, Princeton Univ. Press, Princeton, NJ, 1980.
[15] Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. · Zbl 0332.10018
[16] M. Ram Murty, A motivated introduction to the Langlands program, Advances in number theory (Kingston, ON, 1991) Oxford Sci. Publ., Oxford Univ. Press, New York, 1993, pp. 37 – 66. · Zbl 0806.11054
[17] -, Selberg conjectures and Artin L-functions. II (to appear).
[18] Atle Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno, 1992, pp. 367 – 385. · Zbl 0787.11037
[19] F. Shahidi, On non-vanishing of L-functions, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 462-464. · Zbl 0441.12001
[20] M. F. Vignéras, Facteurs gamma et équations fonctionelles, Lecture Notes in Math., vol. 627, Springer-Verlag, Berlin and New York, 1976. · Zbl 0373.10027
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