Mixed Siegel modular forms and Kuga fiber varieties
type: Item
instance of: scholarly article
title: Mixed Siegel modular forms and Kuga fiber varieties
zbMATH Open document ID: 0795.11021
author: Min Ho Lee
published in: Illinois Journal of Mathematics
publication date: 1994-08-23T00:00:00Z
review text: 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 \(\rho\).
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 14K10
zbMATH DE Number: 622564
zbMATH Keywords: Kuga fiber variety
zbMATH Keywords: mixed Siegel modular forms
zbMATH Keywords: space of holomorphic forms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 622564