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Local to global trace questions and twists of genus one curves. (English) Zbl 1322.11056

The authors consider an elliptic curve \(E\) defined over a number field \(F\) and \(K/F\) of a quadratic extension. They determine necessary and sufficient conditions for a point \(P\in E(F)\), that is a local trace for every completion of \(K/F\), to lie in the image of the global trace map.
In the special case of the quadratic twists of genus one modular curves \(X_0(N)\), where \(N\) is squarefree, the existence of rational points corresponds to the existence of \(\mathbb{Q}\)-curves of degree \(N\) defined over \(K\).
In this paper, the main results have been very well illustrated with several concrete examples.

MSC:

11G05 Elliptic curves over global fields
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References:

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