The non-existence of stable Schottky forms

TL;DR

This paper proves that there are no stable Siegel modular forms vanishing on all moduli spaces of curves, implying that theta series of different quadratic forms can be distinguished by certain curves.

Contribution

It establishes the non-existence of non-trivial stable Schottky forms and explores the intersection properties of moduli spaces in the Satake compactification.

Findings

No stable Schottky forms vanish on all moduli spaces of curves.

The intersection of $M_{g+m}^S$ and $A_g^S$ contains infinitesimal neighborhoods.

Curve period matrices can distinguish between theta series of different quadratic forms.

Abstract

Let $A_g^S$ be the Satake compactification of the moduli space $A_g$ of principally polarized abelian $g$-folds and $M_g^S$ the closure of the image of the moduli space $M_g$ of genus $g$ curves in $A_g$ under the Jacobian morphism. Then $A_g^S$ lies in the boundary of $A_{g+m}^S$ for any $m$. We prove that $M_{g+m}^S$ and $A_g^S$ do not meet transversely in $A_{g+m}^S$, but rather that their intersection contains the $m$th order infinitesimal neighbourhood of $M_g^S$ in $A_g^S$. We deduce that there is no non-trivial stable Siegel modular form that vanishes on $M_g$ for every $g$. In particular, given two inequivalent positive even unimodular quadratic forms $P$ and $Q$, there is a curve whose period matrix distinguishes between the theta series of $P$ and $Q$.