On certain vector valued Siegel modular forms of degree two
Let \(\Gamma_ 2\) be the full Siegel modular group of degree two, \(H_ 2\) the Siegel upper half plane of degree two and \(S_ 2\) the \({\mathbb{C}}\)- vector space of all complex symmetric matrices of size two. A holomorphic Siegel modular form f of type (k,2) and of degree two is an \(S_ 2\) valued holomorphic function on \(H_ 2\) satisfying the equation \[ f((AZ+B) (CZ+D)^{-1})=| CZ+D|^ k (CZ+D) f(Z)^ t(CZ+D) \] for all \(Z\in H_ 2\) and \(M=\left( \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right)\in \Gamma_ 2\). We denote by \(M_ k(\Gamma_ 2)\) the set of (usual scalar valued) holomorphic Siegel modular forms of weight k. For \(f\in M_ k(\Gamma_ 2)\) and \(g\in M_ j(\Gamma_ 2)\), we put \([f,g]=(1/2\pi i)((1/k)g (d/dZ)f-(1/j)g (d/dZ)f).\) Then [f,g]\(\in M_{k+j,2}(\Gamma_ 2)\) [see the final section of \textit{G. Shimura}, Duke Math. J. 44, 365-387 (1977; Zbl 0371.14023)]. Then the author proves the following result. Theorem. For an even integer \(k\geq 0\), we have (as a \({\mathbb{C}}\)-vector space) \[ M_{k,2}(\Gamma_ 2)=M_{k-10}(\Gamma_ 2)[\phi_ 4,\phi_ 6]\quad \oplus \quad M_{k-14}(\Gamma_ 2)[\phi_ 4,\chi_{10}]\quad \oplus \quad M_{k-16}(\Gamma_ 2)[\phi_ 4,\chi_{12}]\quad \oplus \] \[ \oplus \quad V_{k-16}(\Gamma_ 2)[\phi_ 6,\chi_{10}]\quad \oplus \quad V_{k-18}(\Gamma_ 2)[\phi_ 6,\chi_{12}]\quad \oplus \quad W_{k-22}(\Gamma_ 2)[\chi_{10},\chi_{12}] \] where \(V_ k(\Gamma_ 2)=M_ k(\Gamma_ 2)\cap {\mathbb{C}}[\phi_ 6,\chi_{10},\chi_{12}]\) and \(W_ k(\Gamma_ 2)=M_ k(\Gamma_ 2)\cap {\mathbb{C}}[\chi_{10},\chi_{12}].\) [For particular elements of modular forms, see \textit{H. Maass}, Math. Ann. 232, 163-175 (1978; Zbl 0349.10020)]. Some congruences of eigenvalues of Hecke operators are also proved. In the proofs, some properties of real analytic modular forms are used.
congruences of eigenvalues of Hecke operators
scientific article; zbMATH DE number 3910489
scientific article