Cubic surfaces with special periods

@inproceedings{Carlson2011CubicSW,
  title={Cubic surfaces with special periods},
  author={James A. Carlson and Domingo Toledo},
  year={2011}
}
We show that the vector of period ratios of a cubic surface is rational over Q(ω), where ω = exp(2πi/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 K0. The ring of rational endomorphisms of the associated abelian variety is K = K0(ω). 

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On the Abelian fivefolds attached to cubic surfaces

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