Abstract
This paper is a survey on the arithmetic of ℚ-curves: the elliptic curves defined over number fields which are isogenous to all their Galois conjugates. Our purpose is to review some results concerning their basic properties such as: the moduli classification, fields of definition, relationship with abelian varieties of GL2-type, and optimal quotients. Most of the results were separately published before and our aim here is to reinforce a global presentation. What we call “central” Q-curves will constitute the leitmotif.KeywordsElliptic CurveElliptic CurfCohomology ClassAbelian VarietyNumber FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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