Genus two curves with quaternionic multiplication and modular jacobian

Details Genus two curves with quaternionic multiplication and modular jacobian Genus two curves with quaternionic multiplication and modular jacobian

2008-05-09

arxiv.org/abs/0805.1302v1

Mathematics

Number Theory

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

Scientific paper

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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