Abelian surfaces of GL2-type as Jacobians of curves

Abstract

We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian varieties are isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.

To appear in Acta Arithmetica

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• Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity
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