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Fields of definition of -curves

Jordi Quer — 2001

Journal de théorie des nombres de Bordeaux

Let C be a -curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C ' isogenous to C having all the isogenies between its Galois conjugates defined over K , and also the curves C ' isogenous to C defined over a number field K such that the abelian variety Res K / ( C ' / K ) obtained by restriction of scalars is a product of abelian varieties of GL 2 -type.

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