descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 01 Jan 2009Embargo end date: 01 Jan 2008 English Publisher:American Mathematical Society (AMS)Journal:Mathematics of Computation, volume 78, pages 575-575 (issn: 0025-5718,

Authors: González i Rovira, Josep; Guàrdia Rubies, Jordi;

doi: 10.1090/s0025-5718-08-02165-0 , 10.48550/arxiv.0805.1302

arXiv: 0805.1302

handle: 2117/1958

Genus two curves with quaternionic multiplication and modular Jacobian

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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 $A_f$.

15 pages, 2 figures. To appear in Mathematics of Computation

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Keywords

Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry), Abelian varieties of dimension $> 1$, Mathematics - Algebraic Geometry, :14 Algebraic geometry::14H Curves [Classificació AMS], Abelian varieties, Corbes modulars, Genus two curves, FOS: Mathematics, Number Theory (math.NT), Arithmetic aspects of modular and Shimura varieties, Algebraic Geometry (math.AG), 11G10, 11G18, Quaternionic multiplication, quaternionic multiplication, Mathematics - Number Theory, Modular curves, Classificació AMS::14 Algebraic geometry::14H Curves, Varietats abelianes, genus two curves, :11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) [Classificació AMS], modular Abelian surfaces, Quaternions, Modular abelian surfaces

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