# zbMATH — the first resource for mathematics

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)
Full Text: