On symmetric Hilbert modular forms
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)].
scientific article; zbMATH DE number 977857
scientific article