Genus two curves with quaternionic multiplication and modular jacobian

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Title:	Genus two curves with quaternionic multiplication and modular jacobian
Author:	González i Rovira, Josep; Guàrdia Rubies, Jordi
Other authors:	Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. TN - Grup de Recerca en Teoria de Nombres
Abstract:	We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$.
Subject(s):	-Quaternions -Modular curves -Abelian varieties -Genus two curves -Quaternionic multiplication -Modular abelian surfaces -Quaternions -Corbes modulars -Varietats abelianes -Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) -Classificació AMS::14 Algebraic geometry::14H Curves
Rights:	Attribution-NonCommercial-NoDerivs 2.5 Spain http://creativecommons.org/licenses/by-nc-nd/2.5/es/
Document type:	Article
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