arXiv:0907.5263 Geometry of the Siegel modular threefold with paramodular level structure from arXiv Front: math.AG by Chia-Fu Yu In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three standard methods for studying moduli spaces of abelian varieties.
Matthias Schuett has at least one arxiv preprint on modularity of some higher-dimensional varieties.
[arXiv:1212.4308] Modularity of Calabi–Yau varieties: 2011 and beyond from arXiv Front: math.NT by Noriko Yui This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard–Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi-Yau varieties, and (4) the modularity of moduli for families of Calabi-Yau varieties.
http://mathoverflow.net/questions/72951/arithmetic-and-moduli-spaces-of-higher-genus-curves
http://front.math.ucdavis.edu/1110.0106 mentions modularity conjectures for som threefold
nLab page on Modular varieties