Sign changes of Hecke eigenvalues of Siegel cusp forms of genus two. (English) Zbl 1175.11022
Let \(F\) be a nonzero eigenform of integral weight \(k\) for the full Siegel modular group \(\Gamma_2\), with Hecke eigenvalues \(\lambda_n\) for \(n \in \mathbb N\). The result proved in this paper is that if \(F\) is not contained in the Maaßsubspace, then the sequence \((\lambda_n)_{n \in \mathbb N}\) changes sign infinitely often (we note that if \(k\) is odd then the Maaßsubspace is zero). This generalizes similar results of the author and others for subgroups of \(\text{SL}_2(\mathbb R)\) in M. Knopp, W. Kohnen and W. Pribitkin [Ramanujan J. 7, No. 1–3, 269–277 (2003; Zbl 1045.11027)].
The proof given is concise and clear, and relies upon Landau’s theorem as given in E. Landau [Math. Ann. 61, 527–550 (1905; JFM 37.0224.04), the analytic properties of the spinor zeta-function of \(F\) as proved in A. N. Andrianov [Russ. Math. Surv. 29, No. 3, 45–116 (1974); translation from Usp. Mat. Nauk 29, No. 3(177), 43–110 (1974; Zbl 0304.10020)], and a result of Weissauer on the generalized Ramanujan-Petersson conjecture for \(F\), which can be found in [R. Weissauer, Endoscopy for \(\text{GSp}(4)\) and the cohomology of Siegel modular threefolds, Lecture Notes in Mathematics 1968. Berlin: Springer (2009; Zbl 1273.11089)].
The proof given is concise and clear, and relies upon Landau’s theorem as given in E. Landau [Math. Ann. 61, 527–550 (1905; JFM 37.0224.04), the analytic properties of the spinor zeta-function of \(F\) as proved in A. N. Andrianov [Russ. Math. Surv. 29, No. 3, 45–116 (1974); translation from Usp. Mat. Nauk 29, No. 3(177), 43–110 (1974; Zbl 0304.10020)], and a result of Weissauer on the generalized Ramanujan-Petersson conjecture for \(F\), which can be found in [R. Weissauer, Endoscopy for \(\text{GSp}(4)\) and the cohomology of Siegel modular threefolds, Lecture Notes in Mathematics 1968. Berlin: Springer (2009; Zbl 1273.11089)].
Reviewer: Lloyd Kilford (Bristol)
MSC:
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
References:
| [1] | A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus 2, Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43 – 110 (Russian). |
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| [7] | E. Landau, Über einen Satz von Tschebyschef, Math. Ann. 61, 527-550 (1906). · JFM 37.0224.04 |
| [8] | Hans Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. · Zbl 0224.10028 |
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