On special values of standard \(L\)-functions attached to vector valued Siegel modular forms.
Let \(V\) be a vector space of dimension \(n\geq 1\) over \(\mathbb C\) and \(\text{sym}^l(V)\) the \(l\)-th symmetric tensor product of \(V\), \(l\geq 0\). For \(k\geq 0\), let \(f\) be a \(\text{sym}^l(V)\)-valued Siegel modular form of type \(\text{det}^k\otimes \text{sym}^l\) with respect to \(Sp(n,\mathbb Z)\). Suppose \(f\) is a cusp form and an eigenform. The standard \(L\)-function attached to \(f\) is defined by: \[ L(s,f,\underline{\text{St}}):= \prod_p\left\{(1-p^{-s})\prod_{j=1}^n(1-\alpha_j(p)p^{-s})(1-\alpha_j(p)^{-1}p^{-s})\right\}^{-1}, \] where \(p\) runs over all prime numbers and \(\alpha_j(p)\) are the Satake \(p\)-parameters of \(f\). Let \( {\mathbb Q}(f)\) be the extension field of \({\mathbb Q} \) generated by the eigenvalues on \(f\) of the Hecke algebra over \( \mathbb Q\). Suppose that the Fourier coefficients of \(f\) are in \( {\mathbb Q}(f)\). Suppose that \(k\geq 2n+2\), and let \(1\leq m\leq k-n\), \( m\equiv n(\bmod 2)\). If \(m=1\) assume that \(n\equiv 3(\bmod 4) \). Let \[ A(f):=\frac{L(m,f,\underline{\text{St}})}{\pi^{nk+l+m(n+1)-n(n+1)/2}(f,f)}, \] with \((\cdot, \cdot)\) the Petersson scalar product. The author proves that \[ A(f)^\sigma=A(f^\sigma),\quad \forall\sigma \in\text{Aut}(\mathbb C). \] In particular: \[ A(f)\in {\mathbb Q}(f). \]
scientific article; zbMATH DE number 1547129
scientific article