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On elliptic \(K\)-curves. (English) Zbl 1166.11335

Cremona, John (ed.) et al., Modular curves and Abelian varieties. Based on lectures of the conference, Bellaterra, Barcelona, July 15–18, 2002. Basel: Birkhäuser (ISBN 3-7643-6586-2/hbk). Prog. Math. 224, 81-91 (2004).
Summary: For any field \(K\), a \(K\)-curve is an elliptic curve \(E\) defined over some finite separable extension of \(K\) such that every Galois conjugate of \(E\) is isogenous with \(E\) over a separable closure of \(K\). A \(K\)-curve must either have complex multiplication (CM) or be isogenous to a collection of \(K\)-curves parametrized by a \(K\)-rational point on some modular curve \(X^*(N)\). we give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in \(K\)-curves, and in particular in \(\mathbb Q\)-curves, and give some computational examples and open questions.
For the entire collection see [Zbl 1032.11002].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
14G05 Rational points
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