On special values of standard \(L\)-functions attached to vector valued Siegel modular forms.
type: Item
instance of: scholarly article
title: On special values of standard \(L\)-functions attached to vector valued Siegel modular forms.
zbMATH Open document ID: 1184.11018
author: Noritomo Kozima
published in: Kodai Mathematical Journal
publication date: 2001-03-23T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1138044215
review text: 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: \[ A(f)\in {\mathbb Q}(f). \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 1547129
zbMATH Keywords: Siegel modular form
zbMATH Keywords: \(L\)-function
zbMATH Keywords: special values
MaRDI profile type: Publication
OpenAlex ID: W2086368047
cites work: THE MULTIPLICATIVE ARITHMETIC OR SIEGEL MODULAR FORMS
cites work: Q4019377
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cites work: Special values of zeta functions attached to Siegel modular forms
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cites work: On algebraicity of vector valued Siegel modular forms
DOI: 10.2996/KMJ/1138044215
description: scientific article; zbMATH DE number 1547129