Universally irreducible subvarieties of Siegel moduli spaces

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.