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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cubic surfaces with special periods
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by James A. Carlson and Domingo Toledo PDF
Proc. Amer. Math. Soc. 141 (2013), 1947-1962 Request permission

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
  • Email: jcarlson@claymath.org, jxxcarlson@mac.com
  • Domingo Toledo
  • Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
  • Email: toledo@math.utah.edu
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 1947-1962
  • MSC (2010): Primary 14D07, 14K22
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11484-X
  • MathSciNet review: 3034422