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 Izv. RAN. Ser. Mat., 1995, Volume 59, Issue 3, Pages 77–140 (Mi izv23)

Multiplicative arithmetic of theta-series of odd quadratic forms

V. G. Zhuravlev

Abstract: We study the action of the operators of symplectic Hecke rings of arbitrary degree on the theta-series of positive definite quadratic forms in an odd number of variables with vector-valued spherical coefficients corresponding to irreducible representations of the unitary group. We find a correspondence between generators of the Hecke rings and generalized Eichler–Brandt matrices. We apply these results to obtain conditions for linear dependence of theta-series, necessary conditions for lifting automorphic eigenforms on the orthogonal group to Siegel modular eigenforms, and an Euler expansion for symmetric Dirichlet series as a product of local zeta-functions with coefficients computed explicitly in terms of Eichler–Brandt matrices.

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English version:
Izvestiya: Mathematics, 1995, 59:3, 517–578

Bibliographic databases:

MSC: 11E04, 11F27, 11F30, 11F46, 11F66

Citation: V. G. Zhuravlev, “Multiplicative arithmetic of theta-series of odd quadratic forms”, Izv. RAN. Ser. Mat., 59:3 (1995), 77–140; Izv. Math., 59:3 (1995), 517–578

Citation in format AMSBIB
\Bibitem{Zhu95} \by V.~G.~Zhuravlev \paper Multiplicative arithmetic of theta-series of odd quadratic forms \jour Izv. RAN. Ser. Mat. \yr 1995 \vol 59 \issue 3 \pages 77--140 \mathnet{http://mi.mathnet.ru/izv23} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1347079} \zmath{https://zbmath.org/?q=an:0894.11021} \transl \jour Izv. Math. \yr 1995 \vol 59 \issue 3 \pages 517--578 \crossref{https://doi.org/10.1070/IM1995v059n03ABEH000023} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TJ19700004} 

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This publication is cited in the following articles:
1. B. Asch, F. Blij, “Integral quadratic forms and Dirichlet series”, Ramanujan J, 2010
2. Lynne H. Walling, “A formula for the action of Hecke operators on half-integral weight Siegel modular forms and applications”, Journal of Number Theory, 133:5 (2013), 1608
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