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Explicit formulas for local factors in the Euler products for Eisenstein series1)

Published online by Cambridge University Press:  22 January 2016

Paul Feit
Paul Feit*
Affiliation:
Department of Mathematics University of Chicago, Chicago, Illinois60637
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Our objective is to prove that certain Dirichlet series (in our variable q−s), which are defined by infinite sums, can be expressed as a product of an explicit rational function in q−s times an unknown polynomial M in q−s Moreover we show that M(q−s) is 1 if a simple condition is met. The Dirichlet series appear in the Euler products of Fourier coefficients for Eisenstein series. The series discussed below generalize the functions α0(N, q−s) used by Shimura in [12], and the theorem is an extension of Kitaoka’s result [5].

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 37 - 87
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

Footnotes

1)

The work on this paper was partially supported by NSF Grant DMS 8601130.

References

[1] Braun, Hel, Hermitian modular functions, I, II, Ann. of Math. (2), 50, 51 (1949, 1950), 827–855,92–104.Google Scholar
[2] Bump, D., Automorphic Forms on GL(3, R) Springer Verlag, Lecture Notes in Mathematics, 1083, New York, 1984.Google Scholar
[3] Feit, P., Poles and resiudes of Eisenstein series for symplectic and unitary groups, Memoirs of the Amer. Math. Soc., 346 (1986).Google Scholar
[4] Goldman, J. and Rota, G.-C., On the foundations of combinatorical theory IV, Studies in Appl. Math., 49 (1970), 239–258.Google Scholar
[5] Kitaoka, Y., Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J., 95 (1984), 73–84.Google Scholar
[6] Kitaoka, Y., Local densities of quadratic forms and fourier Coefficients of Eisenstein Series, Nagoya Math. J., 103 (1986), 149–160.Google Scholar
[7] Maass, H., Springer Lecture Notes, Vol. 216, 300–318.CrossRefGoogle Scholar
[8] O’Meara, O. T., Introduction to Quadratic Forms, Springer-Verlag, New York, 1973.Google Scholar
[9] Reiner, I., Maximal Orders, Academic Press, New York, 1975.Google Scholar
[10] Shimura, G., Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33–65.CrossRefGoogle Scholar
[11] Shimura, G., Arithmetic of unitary groups, Ann. of Math., 79 (1964), 369–409.CrossRefGoogle Scholar
[12] Shimura, G., On Eisenstein series, Duke Math. J., 50 (1983), 417–476.CrossRefGoogle Scholar
[13] Shimura, G., On Eisenstein series of half-integral weight, Duke Math. J., 52 (1985), 281–314.Google Scholar
[14] Siegel, C. L., Uber die analytische Theorie der quadratischen Formen, Ann. of Math., 36 (1935), 527–606.CrossRefGoogle Scholar
[15] Siegel, C. L., Gesammelte Abh., Vol. II, 125–137.Google Scholar
[16] Sturm, J., The critical values of the zeta functions associated to the symplectic group, Duke Math. J., 48 (1981), 327–350.CrossRefGoogle Scholar
[17] Tsukamoto, T., On the local theory of quaternionic anti-hermitian forms, J. Math. Soc. Japan, 13 (1961), 387–400.CrossRefGoogle Scholar
[18] Weissauer, R., Eisensteinreihen vom Gewicht n+1 zur Siegelschen Modulgruppe n-ten Grades, Math. Ann., 268 (1984), 357–377.Google Scholar