×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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 · doi.org
[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 · doi.org
[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 · doi.org
[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 · doi.org
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.