×

Estimates for eigenvalues of Hecke operators on Siegel cusp forms. (English) Zbl 0858.11027

The author gives new sharp estimates for eigenvalues of Hecke operators on Siegel cusp forms. His main result is Theorem 1. Let \(g\), \(N\) and \(k\) be arbitrary integers with \(g \geq 1\), \(N\geq 3\) and \(k \geq g+1\). Let \(\Gamma_g(N)\) denote the principal congruence subgroup of level \(N\) of \(\text{Sp} (g,\mathbb{Z})\), and let \(S_k (\Gamma_g(N))\) denote the space of the Siegel cusp forms of weight \(k\) on \(\Gamma_g (N)\). Let \(p\) be any prime number with \(p\nmid N\). Let \(\lambda_p\) be any eigenvalue of the Hecke operator \(T_k (p)\) acting on \(S_k (\Gamma_g(N))\). Let \(|\cdot |\) denote any Archimedean absolute value on \(\overline \mathbb{Q}\) that is an extension of the usual Archimedean absolute value on \(\mathbb{Q}\). Put \(w=k-g-1\). Then \(\lambda_p\) enjoys the estimate \[ |\lambda_p |\leq \sum^g_{v=0} p^{e(v)} m(g-v,v) \tag{1} \] where \(e(v)=(v(v+1) + (g-v) (g-v+1) + 2gw)/4\) and \[ \begin{aligned} m(g-v,v) & = \prod^v_{t=1} \bigl((p^{g+1-t}-1)/(p^t-1)\bigr) \quad \text{if } v\geq 1; \\ m(g,0) & = 1 \qquad \text{if } v=0. \end{aligned} \] (For the definition of \(T_k(p)\) acting on \(S_k (\Gamma_g (N))\), see the introduction of this paper and the author’s earlier paper [in: Topics in mathematical analysis, Ser. Pure Math. 11, 371-409 (1989; Zbl 0758.14011)].

MSC:

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

Citations:

Zbl 0758.14011
PDFBibTeX XMLCite
Full Text: DOI Crelle EuDML