Siegel modular forms vanishing on the moduli space of curves
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.
scientific article; zbMATH DE number 3906629
scientific article