Let $X$ be a smooth geometrically irreducible $k$-variety over a number field $k$.
I do not assume that $X$ has a $k$-point.
Is it true that $X$ has $k\_v$-points for almost all places $v$ of $k$?



Rational points over completions of a number field
Without loss of generality $X$ is affine, so embed it in projective space and apply the Bertini Theorem to conclude that $X$ contains a smooth, geometrically integral affine curve $C^{\circ}$ missing $N$ points from its projective completion $C$. Such a guy will remain smooth modulo $v$ for almost all places of $v$ -- this follows immediately from the Jacobian condition -- and now you are reduced to Weil's theorem: you have a family of smooth affine curves $C^{\circ}$ over finite fields $\mathbb{F}\_v$ of fixed genus and missing $N\_v \leq N$ points from its projective completion (in fact $N\_v = N$ for sufficiently large $v$). Now the Weil bounds tell you that the number of $\mathbb{F}\_v$-rational points goes to infinity with $\# \mathbb{F}\_v$. Finally, apply Hensel's Lemma.


