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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Action of Hecke operators on Maass theta series and zeta functions


Author: A. N. Andrianov
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 675-698
MSC (2000): Primary 11F27, 11F46, 11F60, 11F66
Published electronically: June 25, 2008
MathSciNet review: 2381939
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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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Additional Information

A. N. Andrianov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: anandr@pdmi.ras.ru, anatoli.andrianov@gmail.com

DOI: http://dx.doi.org/10.1090/S1061-0022-08-01015-7
PII: S 1061-0022(08)01015-7
Keywords: Harmonic theta series, Hecke operators, Maass theta series, Siegel modular forms, zeta functions of Siegel modular forms
Received by editor(s): April 5, 2007
Published electronically: June 25, 2008
Additional Notes: Supported in part by RFBR (grant no. 05-01-00930).
Dedicated: Dedicated to the centenary of the birth of Dmitriĭ Konstantinovich Faddeev
Article copyright: © Copyright 2008 American Mathematical Society