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Genus two curves with quaternionic multiplication and modular Jacobian


Authors: Josep González and Jordi Guàrdia
Journal: Math. Comp. 78 (2009), 575-589
MSC (2000): Primary 11G10, 11G18
DOI: https://doi.org/10.1090/S0025-5718-08-02165-0
Published electronically: June 18, 2008
MathSciNet review: 2448722
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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$.


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Additional Information

Josep González
Affiliation: Escola Politècnica Superior d’Engenyeria de Vilanova i la Geltrú, Avda Victor Balaguer s/n, 08800 Vilanova i la Geltrú, Spain
Email: josepg@ma4.upc.edu

Jordi Guàrdia
Affiliation: Escola Politècnica Superior d’Engenyeria de Vilanova i la Geltrú, Avda Victor Balaguer s/n, 08800 Vilanova i la Geltrú, Spain
Email: guardia@ma4.upc.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02165-0
Keywords: Genus two curves, quaternionic multiplication, modular abelian surfaces
Received by editor(s): July 10, 2007
Published electronically: June 18, 2008
Additional Notes: The authors were partially supported by MTM2006-15038-C02-02.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.