Local to global trace questions and twists of genus one curves

Abstract:

Let $\mathrm {E}$ be an elliptic curve defined over a number field $\mathrm {F}$ and $\mathrm {K}/\mathrm {F}$ a quadratic extension. For a point $P\in \mathrm {E}(\mathrm {F})$ that is a local trace for every completion of $\mathrm {K}/\mathrm {F}$, we find necessary and sufficient conditions for $P$ to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of $\mathrm {E}$, as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves $X_0(N)$ with squarefree $N$, the existence of rational points corresponds to the existence of $\mathbb {Q}$-curves of degree $N$ defined over $\mathrm {K}$.