Moduli of Abelian Schemes with Real Multiplication

Abstract

Hilbert modular varieties are the moduli spaces of abelian schemes with real multiplication. This interpretation makes it possible to construct models of Hilbert modular varieties over (schemes of) number rings. These models live in a somewhat larger category than that of schemes: they are algebraic stacks. They are however not proper over their base schemes. This is due to the fact that the abelian varieties which we consider can degenerate. Mumford has given a very explicit construction of degenerating abelian schemes, thus generalizing the idea of the Tate curve for elliptic curves. Using this one can construct smooth compactifications of moduli schemes of abelian schemes. The construction was carried out by Rapoport for Hilbert modular varieties over ℤ. Faltings has extended the construction and applied it to the construction of a compactification of A _g , the moduli scheme of principally polarized abelian schemes over ℤ. Faltings has also shown how to obtain a minimal (Baily-Borel) compactification over ℤ.