The Fourier expansion of Eisenstein series for 𝐺𝐿(3,𝑍)

Imai, K.; Terras, A.

doi:10.1090/S0002-9947-1982-0667167-5

The Fourier expansion of Eisenstein series for $\textrm {GL}(3, \textbf {Z})$
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Trans. Amer. Math. Soc. 273 (1982), 679-694 Request permission

Abstract:

The Fourier expansions of Eisenstein series for ${\text {GL}}(3,{\mathbf {Z}})$ are obtained by two methods—one analogous to the classical method used by many number theorists, including Weber, in his derivation of the Kronecker limit formula. The other method is analogous to that used by Siegel to obtain Fourier expansions of Eisenstein series for the Siegel modular group. The expansions involve matrix argument $K$-Bessel functions recently studied by Tom Bengtson. These $K$-Bessel functions are natural generalizations of the ordinary $K$-Bessel function which arise when considering harmonic analysis on the symmetric space of the general linear group using a certain system of coordinates.

References

Introductory lectures on automorphic forms

P. T. Bateman and E. Grosswald, On Epstein’s zeta function, Acta Arith. 9 (1964), 365–373. MR 179141, DOI 10.4064/aa-9-4-365-373

Matrix

-Bessel functions

Atle Selberg and S. Chowla, On Epstein’s zeta-function, J. Reine Angew. Math. 227 (1967), 86–110. MR 215797, DOI 10.1515/crll.1967.227.86
Charles W. Curtis, Representations of finite groups of Lie type, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 5, 721–757. MR 537625, DOI 10.1090/S0273-0979-1979-14648-2
G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1940. MR 0004860
D. R. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111–130. MR 546667
Kaori Imai, Generalization of Hecke’s correspondence to Siegel modular forms, Amer. J. Math. 102 (1980), no. 5, 903–936. MR 590639, DOI 10.2307/2374197
Hervé Jacquet, Ilja Iosifovitch Piatetski-Shapiro, and Joseph Shalika, Automorphic forms on $\textrm {GL}(3)$. I, Ann. of Math. (2) 109 (1979), no. 1, 169–212. MR 519356, DOI 10.2307/1971270
Tomio Kubota, Elementary theory of Eisenstein series, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. MR 0429749
Tomio Kubota, Some results concerning reciprocity law and real analytic automorphic functions, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 382–395. MR 0340221
R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183 (German). MR 31519, DOI 10.1007/BF01329622

Siegel’s modular forms and Dirichlet series

A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Carl Ludwig Siegel, Einführung in die Theorie der Modulfunktionen $n$-ten Grades, Math. Ann. 116 (1939), 617–657 (German). MR 1251, DOI 10.1007/BF01597381
H. M. Stark, On the problem of unique factorization in complex quadratic fields. , Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) Amer. Math. Soc., Providence, R.I., 1967, pp. 41–56. MR 0252350

Fourier analysis on symmetric spaces and applications

Applications of special functions for the general linear group to number theory

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, DOI 10.1090/S0002-9947-1973-0323735-6
A. Terras, Fourier coefficients of Eisenstein series of one complex variable for the special linear group, Trans. Amer. Math. Soc. 205 (1975), 97–114. MR 369267, DOI 10.1090/S0002-9947-1975-0369267-2

Noneuclidean harmonic analysis

24

Audrey Terras, Analysis on positive matrices as it might have occurred to Fourier, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 442–478. MR 654544
Audrey Terras, On automorphic forms for the general linear group, Rocky Mountain J. Math. 12 (1982), no. 1, 123–143. MR 649746, DOI 10.1216/RMJ-1982-12-1-123

Lehrbuch der Algebra

André Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88, Springer-Verlag, Berlin-New York, 1976. MR 0562289