Cubic Surfaces with Special Periods

TL;DR

This paper characterizes when the period ratios of cubic surfaces are rational over a specific field and provides a method to construct such surfaces from certain number fields, linking algebraic geometry and number theory.

Contribution

It establishes a criterion for the rationality of period ratios over $Q(rac{2 extpi i}{3})$ and introduces a construction method for cubic surfaces from totally real quintic fields.

Findings

Period ratios are rational over $Q(rac{2 extpi i}{3})$ iff the abelian variety is isogenous to Fermat elliptic curves.

Construction of cubic surfaces from totally real quintic number fields.

The endomorphism ring of the associated abelian variety is $K_0(rac{2 extpi i}{3})$.

Abstract

We show that the vector of period ratios of a cubic surface is rational over $Q(\omega)$, where $\omega = \exp(2\pi i/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to construct cubic surfaces from a suitable totally real quintic number field $K_0$. The ring of rational endomorphisms of the associated abelian variety is $K = K_0(\omega)$.