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Genus theta series, Hecke operators and the basis problem for Eisenstein series. (English) Zbl 1161.11333

Böcherer, Siegfried (ed.) et al., Automorphic forms and zeta functions. In memory of Tsuneo Arakawa. Proceedings of the conference, Rikkyo University, Tokyo, Japan, September 4–7, 2004. Hackensack, NJ: World Scientific (ISBN 981-256-632-5/hbk). 234-261 (2006).
Summary: We derive explicit formulas for the action of the Hecke operator \(T(p)\) on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of Eisenstein series by genus theta series. We also discuss connections of our results with Kudla’s matching principle for theta integrals.
For the entire collection see [Zbl 1091.11003].

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F25 Hecke-Petersson operators, differential operators (one variable)
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
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