Fourier coefficients of Eisenstein series of one complex variable for the special linear group

Terras, A.

doi:10.1090/S0002-9947-1975-0369267-2

Fourier coefficients of Eisenstein series of one complex variable for the special linear group

Author: A. Terras
Journal: Trans. Amer. Math. Soc. 205 (1975), 97-114
MSC: Primary 10D20; Secondary 10H10
DOI: https://doi.org/10.1090/S0002-9947-1975-0369267-2
MathSciNet review: 0369267
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Abstract: The Eisenstein series in question are generalizations of Epstein's zeta function, whose Fourier expansions generalize the formula of Selberg and Chowla (for the binary quadratic form case of Epstein's zeta function). The expansions are also analogous to Siegel's calculation of the Fourier coefficients of Eisenstein series for the symplectic group. The only ingredients not appearing in Siegel's formula are the Bessel functions of matrix argument studied by Herz. These functions generalize the modified Bessel function of the second kind appearing in the Selberg-Chowla formula.

[1] W. L. Baily, Introductory lectures on automorphic forms, Princeton Univ. Press, Princeton, N. J., 1973.
[2] Klaus Barner, Über die Werte der Ringklassen-𝐿-Funktionen reellquadratischer Zahlkörper an natürlichen Argumentstellen, J. Number Theory 1 (1969), 28–64 (German). MR 238775, https://doi.org/10.1016/0022-314X(69)90024-9
[3] Armand Borel, Introduction to automorphic forms, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 199–210. MR 0207650
[4] Erich Hecke, Mathematische Werke, Herausgegeben im Auftrage der Akademie der Wissenschaften zu Göttingen, Vandenhoeck & Ruprecht, Göttingen, 1959 (German). MR 0104550
[5] Carl S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474–523. MR 69960, https://doi.org/10.2307/1969810
[6] Hervé Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France 95 (1967), 243–309 (French). MR 271275
[7] M. L. Karel, On certain Eisenstein series and their Fourier coefficients, Thesis, University of Chicago, 1972.
[8] Günter Kaufhold, Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades, Math. Ann. 137 (1959), 454–476 (German). MR 121485, https://doi.org/10.1007/BF01360845
[9] Max Koecher, Über Dirichlet-Reihen mit Funktionalgleichung, J. Reine Angew. Math. 192 (1953), 1–23 (German). MR 57907, https://doi.org/10.1515/crll.1953.192.1
[10] Hans Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR 0344198
[11] Carl Ludwig Siegel, Einführung in die Theorie der Modulfunktionen 𝑛-ten Grades, Math. Ann. 116 (1939), 617–657 (German). MR 1251, https://doi.org/10.1007/BF01597381
[12] Carl Ludwig Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2) 45 (1944), 577–622. MR 10574, https://doi.org/10.2307/1969191
[13] -, Lectures on quadratic forms, Tata Institute of Fundamental Research, Bombay, 1957.
[14] Carl Ludwig Siegel, Lectures on advanced analytic number theory, Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23, Tata Institute of Fundamental Research, Bombay, 1965. MR 0262150
[15] Audrey Terras, A generalization of Epstein’s zeta function, Nagoya Math. J. 42 (1971), 173–188. MR 286761
[16] Audrey A. Terras, Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Amer. Math. Soc. 183 (1973), 477–486. MR 323735, https://doi.org/10.1090/S0002-9947-1973-0323735-6
[17] L. C. Tsao, Thesis, University of Chicago, 1972.