Abelian surfaces with anti-holomorphic multiplication

Abstract

For appropriate $N\ge 3$ and $d<0,$ the moduli space of principally polarized abelian surfaces with level $N$ structure and anti-holomorphic multiplication by ${\mathcal{O}}_d$ (the ring of integers in $\mathbb{Q}(\sqrt{d})$) is shown to consist of the real points of a quasi-projective algebraic variety defined over $\mathbb{Q}$, and to coincide with finitely many copies of the quotient of hyperbolic 3-space by the principal congruence subgroup of level $N$ in $\mathbf{SL}(2,{\mathcal{O}}_d).$