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Cubic surfaces with special periods

Authors: James A. Carlson and Domingo Toledo
Journal: Proc. Amer. Math. Soc. 141 (2013), 1947-1962
MSC (2010): Primary 14D07, 14K22
Published electronically: January 15, 2013
MathSciNet review: 3034422
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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.

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Additional Information

James A. Carlson
Affiliation: Clay Mathematics Institute, One Bow Street, Cambridge, Massachusetts 02138
Address at time of publication: 25 Murray Street, Apt. 7G, New York, New York 10007

Domingo Toledo
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Received by editor(s): April 10, 2011
Received by editor(s) in revised form: October 2, 2011
Published electronically: January 15, 2013
Additional Notes: This research was partially supported by NSF Grant DMS-0600816. The first author also gratefully acknowledges the support of the Clay Mathematics Institute and of CIMAT
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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