On symmetric Hilbert modular forms
type: Item
instance of: scholarly article
title: On symmetric Hilbert modular forms
zbMATH Open document ID: 0869.11043
DOI: 10.1007/BF02940795
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1997-09-01T00:00:00Z
review text: Given a symmetric matrix \(P\in\text{GL}_g (\mathbb{Q})\) the associated paramodular group \[ \Gamma(P): =\left(\begin{matrix} 1 & 0 \\ 0 & P \end{matrix} \right) \bigl\{M\in M_{2g} (\mathbb{Z});\;M^tJ_PM =J_P \bigr\} \left(\begin{matrix} 1 & 0 \\ 0 & P \end{matrix} \right)^{-1},\;J_P= \left(\begin{matrix} 0 & P \\ -P & 0 \end{matrix}\right), \] is commensurable with the Siegel modular group of genus \(g\). It is shown that the graded rings of modular forms and Jacobi forms with respect to \(\Gamma(P)\) have a structure similar to the Siegel case. The main application concerns Hilbert modular forms. Given a totally real number field \(K\) of degree \(g>1\) the author constructs a particular \(P=P_K\) and an embedding of the graded ring of symmetric Hilbert modular forms into the graded ring of paramodular forms associated with \(\Gamma(P)\). This generalizes \textit{W. E. Hammond}'s work on modular embeddings in the case of real quadratic number fields [Math. Ann. 200, 25-45 (1973; Zbl 0243.14011)].
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F55
zbMATH DE Number: 977857
zbMATH Keywords: Siegel modular group
zbMATH Keywords: graded rings of modular forms
zbMATH Keywords: Jacobi forms
zbMATH Keywords: symmetric Hilbert modular forms
zbMATH Keywords: paramodular forms
author: Bernhard Runge
reviewed by: Aloys Krieg
MaRDI profile type: Publication
cites work: Fortsetzung von automorphen Funktionen
cites work: Q3963086
cites work: Q3997225
cites work: Bemerkung zu einem Satz von J. Igusa und W. Hammond
cites work: Q4039833
cites work: The Modular Groups of Hilbert and Siegel
cites work: The Hilbert modular surface of a real quadratic field
cites work: Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions
cites work: Q5664012
cites work: Zur Theorie der Siegelschen Modulgruppe
cites work: Q5628266
cites work: Q1838549
cites work: Q4166840
cites work: Su una generazione del gruppo simplettico modulare di Siegel-Conforto \(\Gamma (n,\Delta )\)
cites work: On Siegel modular forms. Part I.
cites work: On Siegel modular forms part II
cites work: Theta functions and Siegel-Jacobi forms
cites work: ON THE GROUPSL2OVER DEDEKIND RINGS OF ARITHMETIC TYPE
description: scientific article; zbMATH DE number 977857