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Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number $d=d(\overline{X})$ with the following property:

For any $k$-form $X$ of $\overline{X}$, the variety $X$ has a $K$-point over some finite field extension $K$ of $k$ of degree $[K:k]\le d$ ?

The answer YES would imply Theorem 2 of my answer to this question.

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No: this fails already when $X=E$ is an elliptic curve and $k=\mathbb{Q}$. This would imply that every element of $H^1(\mathbb{Q},E)$ has order at most $d$, and I'm pretty sure that this cohomology group has elements of arbitrarily large order. Otherwise the Tate-Shafarevich conjecture would be rather trivial, as it is well-known that $Sha(E)[d]$ is finite for every $d$.

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    $\begingroup$ Thank you, Daniel! This shows that my Theorem 2 about principal homogeneous spaces of linear algebraic groups is not trivial.... $\endgroup$
    – Mikhail Borovoi
    Commented May 20, 2016 at 20:12

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