Mixed Siegel modular forms and Kuga fiber varieties
Let \(G\) be a semisimple Lie group whose quotient \(G/K\) by a maximal compact subgroup \(K\) is isomorphic to the Siegel upper half space \({\mathcal H}^ m\) of degree \(n\), and let \(\Gamma\) be a discrete subgroup of \(G\) such that the quotient \(X= \Gamma\setminus {\mathcal H}^ m\) has a structure of a complex manifold. Then both \(G\) and \(Sp(m,\mathbb{R})\) act on \({\mathcal H}^ m\). Let \(\rho: G\to Sp(m,\mathbb{R})\) be a homomorphism and let \(\omega: {\mathcal H}^ m\to {\mathcal H}^ m\) be a holomorphic map such that \(\omega(gz)= \rho(g)\omega(z)\) for all \(g\in G\) and \(z\in {\mathcal H}^ m\). Then the equivariant pair \((\rho, \omega)\) defines a Kuga fiber variety \(\pi: Y_ \omega\to X\) over \(X\) whose fibers are complex tori. In this paper, the author defines mixed Siegel modular forms and shows that the space of holomorphic forms of the highest degree on the fiber space \(Y_ \omega^ n\) of the \(n\)-fold fiber product \(\pi^ n: Y_ \omega^ n\to X\) of the Kuga fiber variety \(\pi: Y_ \omega\to X\) is canonically embedded in the space \({\mathcal M}_{m+1,n} (\Gamma,\omega,\rho)\) of mixed Siegel modular forms of type \((m+1,n)\) associated to \(\Gamma\), \(\omega\) and \(\rho\).
scientific article; zbMATH DE number 622564