A note on Satake parameter of Siegel modular forms of degree 2
Let \(S_k\) be the space of Siegel cusp forms of weight \(k\) of the full Siegel modular group \(Sp(2, \mathbb{Z})\) of degree 2. Then if \(f \in S_k\) is a non-zero common eigenfunction of the Hecke algebra the spinor \(L\)-function attached to \(f\) is defined by \[ \begin{multlined} L(s,f, \text{spin}) \\ = \prod_p \bigl\{(1-\alpha_{0,p} p^{-s})(1-\alpha_{0,p} \alpha_{1,p} p^{-s}) (1- \alpha_{0,p} \alpha_{2,p} p^{-s}) \cdot (1-\alpha_{0,p} \alpha_{1,p} \alpha_{2,p} p^{-s}) \bigr\}^{-1}, \end{multlined} \] where \(p\) runs over all primes and the \(\alpha_{j,p}\)'s are the Satake parameters of \(f\). The cusp form \(f\) is said to satisfy the Ramanujan-Petersson conjecture if the absolute values of the Satake parameters \(\alpha_{1,p}\) and \(\alpha_{2,p}\) of the Euler factors equal 1 for all primes. It is known that for even \(k\) and \(f\) belonging to the Maaß space \(S_k^* \subset S_k\) the form \(f\) does not satisfy this conjecture. By analysing elementary properties of the \(L\)-functions involved the author shows that under various conditions on the zeros of the polynomials attached to the Euler factors of the spinor and standard \(L\)-function by substituting \(p^{-s}\) by the indeterminate \(t\) resp. conditions on the sign of the eigenvalues \(\lambda_f (p)\) of the Hecke operator \(T(p)\) that \(f\) satisfies the Ramanujan-Petersson conjecture.
scientific article; zbMATH DE number 880002