Abelian Varieties with Quaternion Multiplication

2005/07/23 by Ron Donagi, Donagi, Ron, Ron Livné +1
Mathematics · #14H30 #14H37 #14H40 #14J30 #14K30 #14N20 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H30 #msc:14H37 #msc:14H40 #msc:14J30 #msc:14K30 #msc:14N20

paper · pdf · doi:10.48550/arxiv.math/0507493

34 pages. An appendix added for the new version

arxiv created 2005/12/02 · arxiv updated 2009/12/01

Abstract

In this article we use a Prym construction to study low dimensional abelian varieties with an action of the quaternion group. In special cases we describe the Shimura variety parameterizing such abelian varieties, as well as a map to this Shimura variety from a natural parameter space of quaternionic abelian varieties. Our description is based on the moduli of cubic threefolds with nine nodes, which we study in some detail.

Related

• The field of moduli of quaternionic multiplication on abelian varieties
• Rational points on Shimura varieties classifying abelian varieties with quaternionic multiplication
• Abelian varieties with quaternion and complex multiplication
• Shimura curves embedded in Igusa's threefold
• Abelian surfaces with an automorphism and quaternionic multiplication