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Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$ of genus $1$.

There exists a number field $L/K$ such that $X$ has a $L$-rational point. Let $L$ be such a field extension of minimal degree.

Can we bound $[L:\mathbf{Q}]$?

What if we replace number fields by function fields?

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No, for every number field $K$ and $n \in \mathbb{Z}^+$, there exists a genus one curve $C_{/K}$ such that the minimal degree $[L:K]$ of a finite field extension $L$ such that $C$ admits an $L$-rational point is $n$. This is a $2004$ theorem of mine.

Over function fields of characteristic $p > 0$, the same result holds, at least if you take $n$ to be prime to $p$. I think this result follows from the recent (solo) work of (my friend and collaborator) Shahed Sharif. (The result should hold in complete generality, and I believe Shahed and I already know how to prove it. But studying the period-index problem for $p$-torsion classes in characteristic $p$ in full generality gets rather technical, and we have put this aside for a couple of years now...) In any case it is much easier to see that the minimal degree can be arbitrarily large as one varies over all genus one curves $C_{/K}$ even with a fixed Jacobian elliptic curve $E$. For a proof of this -- and actually of a significantly more general result -- see Theorem 11 in these unpublished/unpublishable lecture notes.

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Very nice Crelle paper by the way. –  user18119 Feb 24 '12 at 12:54

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