Introduction

Abstract

Let F be a real quadratic field of discriminant d _F and different ${\mathfrak D}$ _F , and denote by σ ε Gal(F/ℚ) the nontrivial Galois automorphism of F. Associated to F is a Hilbert modular surface M. The algebraic stack M is defined as the moduli space of abelian surfaces A equipped with an action of ${\mathcal O}$ _F , and with an ${\mathcal O}$ _F -linear principal polarization; see Chap. 3 for more details. We refer to such A as ${\mathcal O}$ _F -polarized RM abelian surfaces. The abbreviation RM stands for real multiplication, and the ${\mathcal O}$ _F in ${\mathcal O}$ _F -polarization indicates that the polarization is principal (in much of the text we allow a more general class of polarizations; see Sect. 3.1). It is known that M is regular, flat over Spec(ℤ) of relative dimension two, and smooth over Spec(ℤ[1/d _F ]).