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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11484-X

Published electronically:
January 15, 2013

MathSciNet review:
3034422

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Abstract | References | Similar Articles | Additional Information

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 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 , where 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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11484-X

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.