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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus 2, Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43–110 (Russian). MR 0432552
  • 2. Anatolij N. Andrianov, Quadratic forms and Hecke operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 286, Springer-Verlag, Berlin, 1987. MR 884891
  • 3. A. N. Andrianov, Composition of solutions of quadratic Diophantine equations, Uspekhi Mat. Nauk 46 (1991), no. 2(278), 3–40, 240 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 2, 1–44. MR 1125271, 10.1070/RM1991v046n02ABEH002789
  • 4. A. N. Andrianov, Multiplicative decompositions of integral representations of binary quadratic forms, Algebra i Analiz 5 (1993), no. 1, 81–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 71–95. MR 1220490
  • 5. A. N. Andrianov, Symmetries of harmonic theta functions of integer-valued quadratic forms, Uspekhi Mat. Nauk 50 (1995), no. 4(304), 3–44 (Russian); English transl., Russian Math. Surveys 50 (1995), no. 4, 661–700. MR 1357882, 10.1070/RM1995v050n04ABEH002578
  • 6. A. N. Andrianov, Harmonic theta functions and Hecke operators, Algebra i Analiz 8 (1996), no. 5, 1–31 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 5, 695–720. MR 1428987
  • 7. E. Freitag, Die Wirkung von Heckeoperatoren auf Thetareihen mit harmonischen Koeffizienten, Math. Ann. 258 (1981/82), no. 4, 419–440 (German). MR 650947, 10.1007/BF01453976
  • 8. Eberhard Freitag, Singular modular forms and theta relations, Lecture Notes in Mathematics, vol. 1487, Springer-Verlag, Berlin, 1991. MR 1165941
  • 9. Hans Maass, Konstruktion von Spitzenformen beliebigen Grades mit Hilfe von Thetareihen, Math. Ann. 226 (1977), no. 3, 275–284. MR 0444575
  • 10. Riccardo Salvati Manni and Jaap Top, Cusp forms of weight 2 for the group Γ₂(4,8), Amer. J. Math. 115 (1993), no. 2, 455–486. MR 1216438, 10.2307/2374865
  • 11. Hiroyuki Yoshida, Siegel’s modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193–248. MR 586427, 10.1007/BF01390016

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 11F27, 11F46, 11F60, 11F66

Retrieve articles in all journals with MSC (2000): 11F27, 11F46, 11F60, 11F66


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
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