The ring class field conjecture for Néron–Severi groups of singular Enriques surfaces

Let $Y$ be an Enriques surface whose universal cover $X$ is a singular K3 surface. Let $d<0$ denote the discriminant of $X$, and let $H(d)$ and $H(4d)$ denote the corresponding ring class fields. Ring class field conjecture. $Y$ admits a model over the ring class field $H(d)$ with $\operatorname{NS}(Y)$ defined over $H(4d)$. In examples over $\mathbb{Q}$ with Picard number $20$, this field of definition is observed to be $H(4d)$; the conjecture asserts that the same phenomenon holds for every such Enriques surface.