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On a Space of Some Theta Functions

Published online by Cambridge University Press:  22 January 2016

Yoshiyuki Kitaoka
Yoshiyuki Kitaoka*
Affiliation:
Mathematical Institute, Nagoya Universitya
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In the theory of modular forms there is an interesting problem whether every modular form can be expressed as a linear comination of theta functions. For this Eichler proved in [1] that for a sufficiently large prime q all modular forms of degree — 2m(m = 1,2, · · ·) for Γ0(q) can be represented by linear combinations of theta functios of degree — 2m with level 1 and q. We prove this theorem for q = 2, 3, 5 and 11 by using a theorem of Siegel for q = 2, 3, 5 and a general result of Eichler for q = 11. The former method is shown in Schoeneberg [2].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 42 , March 1971 , pp. 89 - 93
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Eichler, M., Über die Darstellbarkeit von Modulformen durch Thetareihen, J. Reine Angew. Math. 195 (1956), 156–171.Google Scholar
[2] Maass, H., Konstruktion ganzer Modulfomen halbzahliger Dimension mit ϑ-Multiplikatoren in einer und zwei Variabeln, Abh. Math. Sem. Hamburg, 12, (1938), 133–162.CrossRefGoogle Scholar
[3] Schoeneberg, B., Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511–523.CrossRefGoogle Scholar
[4] Siegel, C.L., Über die analytische Theorei der quadratischen Formen, Ann. of Math. 36 (1935), 527–606.Google Scholar