This paper studies the arithmetic of Enriques surfaces whose universal covers are singular \(K3\) surfaces, i.e., \(K3\) surfaces with the maximal Picard number 20. These Enriques surfaces are called singular Enriques surfaces. Singular \(K3\) surfaces are closely related to elliptic curves with complex multiplication (CM). Singular Enriques surfaces share some arithmetic properties with singular \(K3\) surfaces, for instance, the field of definition, as proved in the following theorem.
Theorem. Let \(Y\) be an Enriques surface whose universal cover \(X\) is a singular \(K3\) surface. Let \(d<0\) denote the discriminant of \(X\). Then \(Y\) admits a model over the ring class field \(H(d)\).
The result is known for a singular \(K3\) surface, and the theorem asserts that the same holds for a singular Enriques surface. This is proved by studying Néron-Severi groups of singular \(K3\) surfaces in detail. In other aspects, however, singular Enriques surfaces do behave differently from singular \(K3\) surfaces. For instance, if one considers the Galois action on Néron-Severi groups on singular Enriques surfaces and those on singular \(K3\) surfaces, the fields of definition would be different.
Even if a singular K3 surface \(X\) of discriminant \(d < 0\) admits a model over a smaller field than \(H(d)\), the ring class field \(H(d)\) is preserved through the Galois action on the Néron-Severi group \(\mathrm{NS}(X)\). On the other hand, for singular Enriques surface \(Y\) whose universal cover is \(X\), the Néron-Severi group \(\mathrm{NS}(Y )\) is defined over the ring class field \(H(4d)\), and conjecturally \(Y\) admits a model defined over a ring class field \(H(d)\) with \(\mathrm{NS}(Y )\) defined over \(H(4d)\).