Osaka Journal of Mathematics

Basic vector valued Siegel modular forms of genus two

Eberhard Freitag and Riccardo Salvati Manni

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Abstract

We consider the space $[\Gamma, rr_{0}, v^{r}, \varrho]$ of all vector-valued holomorphic modular forms $f\colon \mathbb{H}_{n} \to \mathcal{Z}$ of transformation type \begin{equation*} f(MZ)=v^{r}(M)\det(CZ+D)^{r_{0}r}\varrho(CZ+D)f(Z). \end{equation*} $\varrho\colon \mathrm{GL}(n, \mathbb{C}) \to \mathrm{GL}(\mathcal{Z})$ is a rational representation on a finite dimensional complex vector space $\mathcal{Z}$. These spaces can be collected in a graded $A(\Gamma)$-module \begin{equation*} \mathcal{M} = \mathcal{M}_{\Gamma}(r_{0}, v, \varrho) := \bigoplus_{r \in \mathbb{Z}}[\Gamma, rr_{0}, v^{r}, \varrho]. \end{equation*} We treat in this paper some special cases in genus 2. The first one is essentially due to Wieber. Here the starting weight is $1/2$, the starting multiplier system is the multiplier system $v_{\Theta}$ and for $\varrho$ we take the second symmetric power of standard representation. Thus we consider a variant of this case and a new example. In this final case the starting weight is $1/2$, the starting multiplier system is the theta multiplier system $v_{\vartheta}$ and for $\varrho$ we take the standard representation. In all these cases we will determine the structure of $\mathcal{M}$.

Article information

Source
Osaka J. Math., Volume 52, Number 3 (2015), 879-895.

Dates
First available in Project Euclid: 17 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1437137623

Mathematical Reviews number (MathSciNet)
MR3370480

Zentralblatt MATH identifier
1330.11031

Subjects
Primary: 11F30: Fourier coefficients of automorphic forms 11F55: Other groups and their modular and automorphic forms (several variables)

Citation

Freitag, Eberhard; Salvati Manni, Riccardo. Basic vector valued Siegel modular forms of genus two. Osaka J. Math. 52 (2015), no. 3, 879--895. https://projecteuclid.org/euclid.ojm/1437137623


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