Genus two curves with quaternionic multiplication and modular Jacobian

González, Josep; Guàrdia, Jordi

doi:10.1090/S0025-5718-08-02165-0

Genus two curves with quaternionic multiplication and modular Jacobian
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by Josep González and Jordi Guàrdia PDF

Math. Comp. 78 (2009), 575-589 Request permission

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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