Action of Hecke operators on Maass theta series and zeta functions

Andrianov, A.

doi:10.1090/S1061-0022-08-01015-7

Action of Hecke operators on Maass theta series and zeta functions
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by A. N. Andrianov
Translated by: the author

St. Petersburg Math. J. 19 (2008), 675-698

DOI: https://doi.org/10.1090/S1061-0022-08-01015-7

Published electronically: June 25, 2008

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Abstract:

The introductory part contains definitions and basic properties of harmonic theta series, Siegel modular forms, and Hecke operators. Then the transformation formulas are recalled, related to the action of modular substitutions and regular Hecke operators on general harmonic theta series, including specialization to the case of Maass theta series. The following new results are obtained: construction of infinite sequences of eigenfunctions for all regular Hecke operators on spaces of Maass theta series; in the case of Maass theta series of genus $2$, all the eigenfunctions are constructed and the corresponding Andrianov zeta functions are expressed in the form of products of two $L$-functions of the relevant imaginary quadratic rings. The proofs are based on a combination of explicit formulas for the action of Hecke operators on theta series with Gauss composition of binary quadratic forms.

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