×

zbMATH — the first resource for mathematics

Eisenstein group cocycles for \(\text{GL}_ n\) and values of \(L\)- functions. (English) Zbl 0809.11029
Let \({\mathfrak b}\) and \({\mathfrak f}\) be two integral ideals in a number field \(F\) and denote by \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s\in\mathbb{C})\) the zeta function of the ray class of \({\mathfrak b}\) modulo \({\mathfrak f}\). By work of Klingen-Siegel the special values \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s=0,- 1, \dots)\) are known to be rational. In his paper the author gives an interpretation of these special values in terms of the cohomology of the group \(\Gamma= \text{GL}_ n (\mathbb{Z})\) \((n=\deg(F))\). More precisely, a so-called Eisenstein cocycle \(\psi\) on \(\Gamma\) is constructed which represents a non-trivial class in \(H^{n-1} (\Gamma, M)\) where \(M\) is a certain function space. Restricting \(\psi\) to the group \(U\) of totally positive units in \(F\) (embedded regularly in \(\Gamma\)) and evaluating functions in \(M\) on \(U\)-invariant points gives classes in \(H^{n-1} (U,\mathbb{C})\); among those certain rational classes when evaluated on a fundamental cocycle in \(H_{n-1} (U,\mathbb{Z})\), give the values \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s=0,-1,\dots)\).

MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Ar] Ash, A., Rudolph, L.: The modular symbol and continued fractions in higher dimensions. Invent. Math.55, 241-250 (1979) · Zbl 0426.10023
[2] [AW] Atiyah, M.F., Wall, C.T.C.: Cohomology of groups. In: Cassels, J.W.S., Fröhlich, A. (eds.) Algebraic number theory, pp. 94-115 London: Academic Press 1967
[3] [Co] Colmez, P.: Algébricité de valeurs spéciales le fonctions L. Invent. Math.95, 161-205 (1989) · Zbl 0666.12008
[4] [Eis] Eisenstein, G.: Genaue Untersuchung der unendlichen Doppelproducte, aus denen die elliptischen Functionen als Quotienten zusammengesetzt sind. J. Reine Angew. Math.35, 153-274 (1847); In: Mathematische Werke I, pp. 357-478. New York: Chelsea 1975 · ERAM 035.0987cj
[5] [HR] Hua, L.K., Reiner, I.: On the generators of the symplectic group. Trans. Am. Math. Soc.65, 415-426 (1949) · Zbl 0034.30503
[6] [Hu] Hurwitz, A.: Über die Anzahl der Klassen positiver ternärer quadratischer Formen von gegebener Determinante. Math. Ann.88, 26-52 (1923) · JFM 48.1164.04
[7] [La] Lang, S.: Algebraic Number Theory, Berlin: Heidelberg, New York: Springer 1986 · Zbl 0601.12001
[8] [Ra] Rademacher, H., Grosswald, E.: Dedekind Sums. (Carus Math. Monogr., vol. 16) Washington: MAA 1972 · Zbl 0251.10020
[9] [Sch] Schellbach, K.H.: Die einfachsten periodischen Functionen. J. Reine Angew. Math. 1b (XLVIII), 207-236 (1854) · ERAM 048.1287cj
[10] [Sc1] Sczech, R.: Eisenstein group cocycles for GL2? and values of L-functions in real quadratic fields. Comment. Math. Helv.67, 363-382 (1992) · Zbl 0776.11021
[11] [Sc2] Sczech, R.: Cusps on Hilbert modular varieties and values of L-functions. In: Hashimoto, V. Namikawa, Y. (eds) Automorphic Forms and Geometry of Algebraic Varieties. (Adv. Stud. Pure Math.,15, vol. pp. 29-40) Amsterdam: North-Holland and Tokyo: Kinokuniya 1989 · Zbl 0703.11025
[12] [Sh] Shintani, T.: On evaluation of zeta functions of totally real algebraic number fields at non-positive integers. J. Fac. Sci. Univ. Tokyo23, 393-417 (1976) · Zbl 0349.12007
[13] [Si1] Siegel, C.L.: Über die Fourierschen Koeffizienten von Modulformen. In: Gesammelte Abhandlungen IV. pp. 98-139. Berlin Heidelberg New York: Springer 1979
[14] [Si2] Siegel, C.L.: Zur Summation von L-Reihen. In: Gesammelte Abhandlungen IV, pp. 305-328. Berlin Heidelberg New York: Springer 1979
[15] [W] Weil, A.: Elliptic functions according to Eisenstein and Kronecker. Berlin, Heidelberg New York: Springer 1976 · Zbl 0318.33004
[16] [We] Weselmann, U.: An elementary identity in the theory of Hecke L-functions. Invent. Math.95, 207-214 (1989) · Zbl 0666.12007
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.