×

Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series. (English) Zbl 1275.11080

Summary: We evaluate the action of Hecke operators on Siegel Eisenstein series of degree 2, square-free level \(\mathcal N\) and arbitrary character \(\chi \), without using knowledge of their Fourier coefficients. From this we construct a basis of simultaneous eigenforms for the full Hecke algebra, and we compute their eigenvalues. As well, we obtain Hecke relations among the Eisenstein series. Using these Hecke relations, we discuss how to generate the Fourier series of Eisenstein series in a basis from the Fourier series of one basis element.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F60 Hecke-Petersson operators, differential operators (several variables)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Böcherer, S., Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen, Manuscripta Math., 45, 273-288 (1984) · Zbl 0533.10023
[2] Choie, Y.; Kohnen, W., Fourier coefficients of Siegel-Eisenstein series of odd genus, J. Math. Anal. Appl., 374, 1-7 (2011) · Zbl 1226.11063
[3] Freitag, E., Siegel Eisenstein series of arbitrary level and theta series, Abh. Math. Semin. Univ. Hambg., 66, 229-247 (1996) · Zbl 0870.11028
[4] Hafner, J. L.; Walling, L. H., Explicit action of Hecke operators on Siegel modular forms, J. Number Theory, 93, 34-57 (2002) · Zbl 1044.11034
[5] Katsurada, H., An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree 3, Nagoya Math. J., 146, 199-223 (1997) · Zbl 0882.11026
[6] Katsurada, H., An explicit formula for Siegel series, Amer. J. Math., 121, 2, 415-452 (1999) · Zbl 1002.11039
[7] Kohnen, W., Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math. Ann., 322, 787-809 (2002) · Zbl 1004.11020
[8] Maass, H., Die Fourierkoeffizienten der Eisensteinreihen zweiten Grades, Mat.-Fys. Medd. Danske Vid. Selsk., 34, 7 (1964), 25 pp · Zbl 0132.06402
[9] Maass, H., Über die Fourierkoeffizienten der Eisensteinreihen zweiten Grades, Mat.-Fys. Medd. Danske Vid. Selsk., 38, 14 (1972), 13 pp · Zbl 0244.10023
[10] Mizuno, Y., An explicit arithmetic formula for the Fourier coefficients of Siegel-Eisenstein series of degree two and square-free odd levels, Math. Z., 263, 837-860 (2009) · Zbl 1234.11057
[11] S. Takemori, \(p\); S. Takemori, \(p\)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.