Abstract
Let F be a positive-definite integral even matrix of even order. For an arbitrary prime number p and natural n one obtains an explicit expression for the image of the theta series of genus n of the matrix F, under the action of an irregular Hecke operator with index p, in the form of a linear combination of theta series.
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Literature cited
A. N. Andrianov and G. N. Maloletkin, “The behavior of theta-series of genus n under modular substitutions,” Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 2, 243–258 (1975).
J. W. S. Cassels, Rational Quadratic Forms, Academic Press, London (1978).
A. Ogg, Modular Forms and Dirichlet Series, Benjamin, New York (1969).
A. N. Andrianov, “Action of Hecke opertor T(p) on theta-series,” Math. Ann.,247, No. 3, 245–254 (1981).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 68–71, 1985.
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Evdokimov, S.A. Action of the irregular Hecke operator of index p on the theta-series of a quadratic form. J Math Sci 38, 2078–2081 (1987). https://doi.org/10.1007/BF01474441
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DOI: https://doi.org/10.1007/BF01474441