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Action of Hecke operators on theta-functions with rational characteristics


Author: A. N. Andrianov
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), nomer 5.
Journal: St. Petersburg Math. J. 15 (2004), 715-732
MSC (2000): Primary 11F27, 11F46, 11F60, 11F66
DOI: https://doi.org/10.1090/S1061-0022-04-00828-3
Published electronically: August 2, 2004
MathSciNet review: 2068791
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Abstract | References | Similar Articles | Additional Information

Abstract: The explicit formulas for the transformation of theta-functions of integral positive definite quadratic forms under the action of regular Hecke operators, obtained in the author's earlier paper (1996), are converted to transformation formulas for the theta-functions with rational characteristics (the theta-series) viewed as Siegel modular forms. As applications, sequences of invariant subspaces and eigenfunctions for all regular Hecke operators on spaces of theta-series are constructed.


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

  • 1. A. N. Andrianov, Multiplicative arithmetic of Siegel's modular forms, Uspekhi Mat. Nauk 34 (1979), no. 1, 67-135; English transl., Russian Math. Surveys 34 (1979), no. 1, 75-148. MR 0525651 (80f:10032)
  • 2. -, Quadratic forms and Hecke operators, Grundlehren Math. Wiss., vol. 286, Springer-Verlag, Berlin-New York, 1987. MR 0884891 (88g:11028)
  • 3. -, Composition of solutions of quadratic Diophantine equations, Uspekhi Mat. Nauk 46 (1991), no. 2, 3-40; English transl., Russian Math. Surveys 46 (1991), no. 2, 1-44. MR 1125271 (93e:11049)
  • 4. -, Factorizations of integral representations of binary quadratic forms, Algebra i Analiz 5 (1993), no. 1, 81-108; English transl., St. Petersburg Math. J. 5 (1994), no. 1, 71-95. MR 1220490 (94h:11034)
  • 5. -, Symmetries of harmonic theta functions of integer-valued quadratic forms, Uspekhi Mat. Nauk 50 (1995), no. 4, 3-44; English transl., Russian Math. Surveys 50 (1995), no. 4, 661-700. MR 1357882 (96i:11041)
  • 6. -, Harmonic theta functions and Hecke operators, Algebra i Analiz 8 (1996), no. 5, 1-31; English transl., St. Petersburg Math. J. 8 (1997), no. 5, 695-720. MR 1428987 (98a:11053)
  • 7. -, Maass theta-series and Hecke operators, Preprint Series no. 48, Max-Planck-Inst. Math., Bonn, 2002.
  • 8. R. Salvati-Manni and J. Top, Cusp forms of weight $2$ for the group $\Gamma_2(4,8)$, Amer. J. Math. 115 (1993), no. 2, 455-486. MR 1216438 (94e:11050)
  • 9. H. Yoshida, Siegel's modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193-248. MR 0586427 (81m:10051)

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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

DOI: https://doi.org/10.1090/S1061-0022-04-00828-3
Keywords: Hecke operators, Siegel modular forms, theta functions of quadratic forms, zeta functions of modular forms
Received by editor(s): April 23, 2003
Published electronically: August 2, 2004
Additional Notes: Supported in part by RFBR (grant no. 02-01-00087) and by SFB 478 “Geometrische Structuren in der Mathematik”, Westfälische Wilhelms-Universität Münster.
Article copyright: © Copyright 2004 American Mathematical Society

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