**Background.**
Let $\mathbb H= \lbrace z \in \mathbb C \; | \; Im(z)>0 \rbrace$ be the upper half-plane, and let
$\Gamma \subset PSL(2,\mathbb R)\times PSL(2,\mathbb R)$ be an irreducible lattice. More concretely $\Gamma$ is a discrete subgroup non-commensurable with a product $\Gamma_1 \times \Gamma_2$ and such that the quotient of $\mathbb H \times \mathbb H$ by $\Gamma$ has finite volume.

Examples of such irreducible lattices come from the natural embedding of $PSL(2,\mathcal O_K) \to PSL(2,\mathbb R) \times PSL(2,\mathbb R)$ where $K$ is any totally real quadratic number field and the embedding is given by $A \mapsto (\sigma_1(A),\sigma_2(A))$ with $\sigma_1,\sigma_2$ being the two distinct embeddings of $K$ into $\mathbb R$.

Let's call the quotient of $\mathbb H \times \mathbb H$ by an irreducible lattice a Hilbert modular surface when the quotient is a quasi-projective surface, even if this is not the standard terminology. Usually the term Hilbert modular surface is reserved to the quotient of $\mathbb H \times \mathbb H$ by certain sublattices of $PSL(2,\mathcal O_K)$.

Back in the seventies and eighties there was some activity around the determination of totally real number fields $K$ for which $X(K)$, a desingularization of a compactification of $\mathbb H\times \mathbb H/PSL(2,\mathcal O_K)$, has Kodaira dimension smaller than two. Much of the research on the subject is summarized on van der Geer's book Hilbert Modular Surfaces. There, if I remember correctly, one can find an exhaustive (and very short) list of number fields for which $X(K)$ is rational.

**Question.** Let $\mathcal S$ be the set of conjugacy classes of
irreducible lattices $\Gamma \subset
PSL(2,\mathbb R)\times PSL(2,\mathbb
R)$ for which the quotient $\mathbb H\times \mathbb
H/ \Gamma$ is rational. Is $\mathcal S$ finite? In other words, is the number of rational Hilbert modular surfaces finite ?