Universally irreducible subvarieties of Siegel moduli spaces

TL;DR

This paper studies the conditions under which subvarieties of Siegel moduli spaces are universally irreducible, and explores the factorization of Siegel modular forms into irreducible components, with implications for special theta series and Schottky forms.

Contribution

It provides new criteria for universal irreducibility of subvarieties and demonstrates that Siegel modular forms can be factored into finitely many irreducible functions for genus g ≥ 3.

Findings

Complete intersection subvarieties of small codimension are universally irreducible under certain conditions.

Every Siegel modular form for g ≥ 3 factors into finitely many irreducible analytic functions.

Special cases include singular theta series of weight 1/2 and Schottky forms.

Abstract

A subvariety of a quasi-projective complex variety $X$ is called ``universally irreducible'' if its preimage inside the universal cover of $X$ is irreducible. In this paper we investigate sufficient conditions for universal irreducibility. We consider in detail complete intersection subvarieties of small codimension inside Siegel moduli spaces of any finite level. Moreover we show that, for $g\geq 3$, every Siegel modular form is the product of finitely many irreducible analytic functions on the Siegel upper half-space $\mathbb{H}_g$. We also discuss the special case of singular theta series of weight $\frac{1}{2}$ and of Schottky forms.