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On the Hecke operator \(U(p)\) (with an appendix by Ralf Schmidt). (English) Zbl 1114.11044
Let \(\Gamma_0(N)\) denote the Hecke subgroup of level \(N\) of Sp\((n;\mathbb{Z})\). Given a prime \(p, p| N\), the author considers the Hecke operator \(U(p)\), which maps the space of Siegel modular forms \([\Gamma_0 (N), k]\) into \([\Gamma_0 (N), k]\) resp. \([\Gamma_0 (N/p), k]\) if \(p^2| N\). It is shown that \(U(p)\) is injective by studying the double coset \(\Gamma_0(p) J\Gamma_0(p)\) in the Hecke algebra associated with \((\Gamma_0 (p), \text{Sp}(n; \mathbb{Z}))\). Moreover the case of nontrivial nebentypus \(\chi\) is included. The appendix contains a different approach using the structure theory of Iwahori-Hecke algebras.
Reviewer: A. Krieg (Aachen)

MSC:
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F60 Hecke-Petersson operators, differential operators (several variables)
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