Hilbert modular surface (awaiting review)

A Hilbert modular surface is a nice algebraic surface that parametrizes abelian surfaces with real multiplication and equipped with level structure.

More precisely, let $F$ be a real quadratic field. Let $\Gamma < \GL_2^+(F)$ be a congruence subgroup. Then $\Gamma$ is a discrete group acting properly by isometries on the product $\overline{\mathfrak{h}}^2$ of completed upper half-planes.

The quotient $\overline{Y}(\Gamma)(\C) := \Gamma \backslash \mathfrak{h}^2$ is the set of complex points of a projective surface $Y(\Gamma)$, canonically defined over a number field $E$ (its reflex field). A Hilbert modular surface is the minimal desingularization $X(\Gamma)$ of $Y(\Gamma)$. In particular, $X(\Gamma)$ is smooth, projective, and geometrically integral.