Cubic surfaces with special periods

Abstract:

We study special values of the point in the unit ball (period) associated to a cubic surface. We show that this point has coordinates in $\mathbb {Q}(\sqrt {-3})$ if and only if the abelian variety associated to the surface is isogenous to the product of five Fermat elliptic curves. The proof uses an explicit formula for the embedding of the ball in the Siegel upper half plane. We give explicit constructions of abelian varieties with complex multiplication by fields of the form $K_0(\sqrt {-3})$, where $K_0$ is a totally real quintic field, which arise from smooth cubic surfaces. We include Sage code for finding such fields and conclude with a list of related problems.