TITLE: varieties with points in number fields
QUESTION [37 upvotes]: Let $V$ be a projective variety, defined over $\mathbb{Q}$. Suppose that for every number field $K \neq \mathbb{Q}$, there is a $K$-point of $V$. Does it follow that $V$ has a $\mathbb{Q}$-point?
More generally, what are the restrictions on the possible sets $S$ of number fields $K \supset k$ such that for some projective variety $V$ defined over $k$, we have that $V$ has a $K$-point if and only if $K \in S$? Note that there has to be some nontrivial restriction here, since there are only countably many projective varieties defined over number fields, but uncountably many sets of number fields closed under extensions.

REPLY [8 votes]: A simple case where the answer to your question is positive is provided by Heegner's Lemma, an observation made by Heegner in his solution of the class number 1 problem and extracted from his proof by Birch: if $y^2 = f_4(x)$, where $f_4$ is a quartic polynomial whose leading coefficient is not a square, has a solution in a number field of odd degree, then it has a rational point. This lemma has a cohomological interpretation, where the claim essentially boils down to the observation that if $a^n$ is a square for some odd $n$, then so is $a$. 
I expect that there are other instances of Heegner's lemma for other types of curves, but I don't know anything specific.