Abstract
In Kim [7], we studied an Eisenstein series on quaternion half-space of degree 2. By calculating the Siegel series using the method of Karel [5], we obtained the analytic continuation and functional equation of the Eisenstein series. In this note we study an Eisenstein series on quaternion half-space of degreen. By calculating the Siegel series in an analogous way as in Shimura [15] and Kitaoka [8], we obtain singular modular forms of weightk, k<2n and 4/k. Furthermore, we obtain the analytic continuation and functional equation of the Eisenstein series.
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References
W. L. Baily Jr.,An exceptional arithmetic group and its Eisenstein series, Ann. of Math.91 (1970), 512–549
W. L. Baily, Jr.,On Hensel's Lemma and exponential sums, in Global Analysis, a volume of assorted articles dedicated to K. Kodaira, University of Tokyo Press, 1969
W. L. Baily Jr.,Introductory Lectures on Automorphic Forms, Princeton University Press, 1973
J. Goldman and G.-C. Rota,On the foundations of combinatorial theory IV, Studies in Appl. Math49 (1970), 239–258
M. Karel,Fourier coefficients of certain Eisenstein series, Ann. of Math.99 (1974), 176–202
G. Kauflold,Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktionen 2. Grades, Math. Annalen137 (1959), 454–476
H. Kim,Eisenstein series on quaternion half-space, to appear in Manuscripta mathematica (1992)
Y. Kitaoka,Dirichlet series in the theory of Siegel modular forms, Nagoya Math. J.95 (1984), 73–84
A. Krieg,Modular Forms on Half-Spaces of Quaternions, Lecture Notes in Mathematics 1143, Springer-Verlag, 1985
—,The Elementary Divisor Theory over the Hurwitz Order of Integral Quaternions, Linear and Multilinear Algebra21 (1987), 325–344
—,Eisenstein series on real, complex and quaternionic half-spaces. Pacific Journal of Math.133, no 2 (1988), 315–354
—,The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions, Proc. Indian Acad. Sci. (Math. Sci.)97 (1987), 201–229
H. Maass,Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Math. 216, Springer-Verlag, 1971
G. Shimura,Confluent Hypergeometric Functions on Tube Domains, Math. Annalen260 (1982), 269–302
—,On Eisenstein Series, Duke Math. J.50 (1983), 417–476
—,Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan15, No 1 (1963), 33–65
C. L. Siegel,Einfuhrung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann.116 (1939), 617–657
T. Tamagawa,On the ζ-functions of a division algebra, Ann. of Math.77, No. 2 (1963), 387–405
L. C. Tsao,The rationality of the Fourier coefficients of certain Eisenstein series in Tube domains (I), Composition Mathematica32 (1976), 225–291
S. T. Wong,The meromorphic continuation and functional equations of cuspidal Eisenstein series for maximal cuspidal groups, Memoirs of the Amer. Math. Soc., Number 423 (1990)
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Kim, H.H. Eisenstein series on quaternion half-space of degree N. Manuscripta Math 77, 215–235 (1992). https://doi.org/10.1007/BF02567055
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DOI: https://doi.org/10.1007/BF02567055