Cubic Surfaces with Special Periods

Details Cubic Surfaces with Special Periods Cubic Surfaces with Special Periods

2011-04-10

arxiv.org/abs/1104.1782v3

Mathematics

Algebraic Geometry

Implemented the referees thoughtful comments and suggestions. Thankyou!

Scientific paper

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 $K = K_0(\omega)$.

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