Abstract
Let K be a number field, Galois over ℚ. A ℚ-curve over K is an elliptic curve over K which is isogenous to all its Galois conjugates. The current interest in ℚ-curves, it is fair to say, began with Ribet’s observation [27] that an elliptic curve over ℚ admitting a dominant morphism from X 1 (N) must be a ℚ-curve. It is then natural to conjecture that, in fact, all ℚ-curves are covered by modular curves. More generally, one might ask: from our rich storehouse of theorems about elliptic curves over ℚ, which ones generalize to ℚ-curves?KeywordsModular FormElliptic CurveElliptic CurfAbelian VarietyCusp FormThese 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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