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?
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Ellenberg, J.S. (2004). ℚ-Curves and Galois Representations. In: Cremona, J.E., Lario, JC., Quer, J., Ribet, K.A. (eds) Modular Curves and Abelian Varieties. Progress in Mathematics, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7919-4_7
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DOI: https://doi.org/10.1007/978-3-0348-7919-4_7
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