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Let be a -curve with no complex multiplication. In this note we characterize the number fields such that there is a curve isogenous to having all the isogenies between its Galois conjugates defined over , and also the curves isogenous to defined over a number field such that the abelian variety Res obtained by restriction of scalars is a product of abelian varieties of GL-type.
In [] we introduced the concept of strongly modular abelian variety. This note contains some remarks and examples of this kind of varieties, especially for the case of Jacobian surfaces, that complement the results of [].
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