Cubic surfaces with special periods

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 Q(−3)\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 use...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2013-06, Vol.141 (6), p.1947-1962
Main Authors: Carlson, James A., Toledo, Domingo
Format: Article
Language:English
Subjects:
Research article
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 Q(−3)\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 K0(−3)K_0(\sqrt {-3}), where K0K_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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-2013-11484-X