Galois \(\epsilon\)-factors modulo roots of unity. (English) Zbl 0557.12011

Let F be a locally compact non-Archimedean field, \(\bar F\) a separable algebraic closure of F, \(G_ F\) the Galois group of \(\bar F\) over F, and p the residue characteristic of F. We fix a non-trivial character \(\psi\) of the additive group F, and on this group we take the Haar measure dx which is self-dual for \(\psi\). For every continuous finite-dimensional complex representation \(\sigma\) of the group \(G_ F\), Langlands and Deligne have proved the existence of the local constant of functional equations of L-functions attached to \(\sigma\),\(\psi\) and dx, also known as the \(\epsilon\)-factor of \(\sigma\). We consider \(\epsilon\) (\(\sigma\),\(\psi\),dx) as a function \(\epsilon\) (\(\sigma)\) of the complex parameter s and write \(\epsilon\) ’(\(\sigma)\) for its value at \(s=1/2\). In view of the local Langlands conjecture it is of foremost importance to understand more about the \(\epsilon\)-factors. P. Deligne and the author have investigated the \(\epsilon\)-factor \(\epsilon\) (\(\sigma)\) when \(\sigma\) is twisted by another less ramified representation [ibid. 64, 89-118 (1981; Zbl 0442.12012)]. In particular they showed that this behaviour is remarkably simple if one considers the values of \(\epsilon\) (\(\sigma)\) only modulo the group \(\mu\) of roots of unity in \({\mathbb{C}}^{\times}\) of p-power order.
In the present paper an easy formula (involving a quadratic Gauss sum) is given for \(\epsilon\) ’(\(\sigma)\) modulo \(\mu\), in a number of interesting cases, notably when \(\sigma\) is nonminimal, or primitive, or irreducible of degree p with Artin exponent prime to p. This plays a crucial role in the proof of the Langlands correspondence for GL(p,F) (p odd) by Kutzko, Moy and the author.


11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
Full Text: DOI EuDML


[1] [BF] Bushnell, C.J., Fröhlich, A.: Gauss sums andp-adie division algebras. Lecture Notes in Mathematics no 987, Berlin-Heidelberg-New York: Springer 1983
[2] [De] Deligne, P.: Les constantes des équations fonctionnelles des fonctionsL in Modular functions of one variable II. Lect. Notes in Math. no 349, Berlin-Heidelberg-New York: Springer 1973
[3] [DH] Deligne, P., Henniart, G.: Sur la variation, par torsion, des constantes locales d’équations fonctionnelles de fonctionsL. Invent. Math.64, 89-118 (1981) · doi:10.1007/BF01393935
[4] [GK] Gérardin, P., Kutzko, Ph.: Facteurs locaux pourGL (2). Ann. Scient E. N. S., t.13, 349-384 (1980)
[5] [He 1] Henniart, G.: Représentations du groupe de Weil d’un corps local. L’Enseign. Math. t.26, 155-172 (1980) · Zbl 0452.12006
[6] [He 2] henniart, G.: La conjecture de Langlands locale pourGL (n),in Journées arithmétiques de Metz (Sept. 1981), Astérisque no 94, pp. 67-85 (1982)
[7] [He 3] Henniart, G.: La conjecture de Langlands locale pourGL (3), Thèse d’Etat, Prépublications Univ. Paris-SUD, May 1983, (to appear in Mémoires de la S.M.F.)
[8] [He 4] Henniart, G.: La conjecture de Langlands locale pourGL (p), Séminaire de Théorie des Nombres de Bordeaux. Année 1982-83, exp. no 28 (to appear)
[9] [He 5] Henniart, G.: Calculs de facteurs ? locaux, Note to the C.R.A.S. Paris (to appear) · Zbl 0564.12018
[10] [KM1] Kutzko, Ph., Moy, A.: On the local Langlands conjecture in prime dimension. B. A. M. S. vol.9, no 3, pp. 323-325 (Nov. 1983) · Zbl 0525.12014 · doi:10.1090/S0273-0979-1983-15192-3
[11] [KM 2] Kutzko, Ph. Moy, A.: On the local Langlands conjecture in prime dimension, preprint, April 1984
[12] [Mo] Moy, A.: Local constants and the tame Langlands correspondence, Ph. D. Dissertation, Chicago (1982)
[13] [Se] Serre, J.-P.: Conducteurs d’Artin des caractères réels. Invent. Math.14, 173-183 (1971) · Zbl 0229.13006 · doi:10.1007/BF01418887
[14] [Ta] Tate, J.: Number-theoretic background,in Automorphic forms, representations andL-functions, PSPM, vol33, part II. A.M.S., Providence pp. 3-26 (1979)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.