Siegel modular forms vanishing on the moduli space of curves
type: Item
instance of: scholarly article
title: Siegel modular forms vanishing on the moduli space of curves
zbMATH Open document ID: 0568.14015
DOI: 10.1007/BF01388598
author: Bert van Geemen
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143176
review text: Let \(M_ g\) be the moduli space of curves of genus g. If \(A_ g\) is the space of principally polarized d-dimensional abelian varieties then there is a natural morphism \(j: M_ g\to A_ g\) defined by sending a curve to its jacobian. It is a classical problem to give a set of equations for the closure \(J_ g\) of the image of the morphism j. When \(g=4\) this problem was solved by Igusa utilizing a result of Schottky. In this paper the author extends Schottky's result to \(g>4\), and in so doing makes a major contribution toward solving the general problem. To be more precise, let \(\bar A_ g\) denote the Satake compactification of \(A_ g\), and let \(\bar J_ g\) denote the closure of \(J_ g\) in \(\bar A_ g\). The author defines an ideal \(S(\Gamma_ g)\) of the graded ring \(A(\Gamma_ g)\) generated by Siegel modular forms on \(\Gamma_ g=Sp(2g,{\mathbb{Z}})\). He then carries out an induction argument that shows that \(S(\Gamma_ g)\) cuts out a subset of \(\bar A_ g\) containing \(\bar J_ g\) as an irreducible component.
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 3906629
zbMATH Keywords: Torelli theorem
zbMATH Keywords: Schottky problem
zbMATH Keywords: theta constants
zbMATH Keywords: moduli space of curves
zbMATH Keywords: Satake compactification
zbMATH Keywords: Siegel modular forms
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3906629