  For every number field $k$, we construct an affine algebraic surface $X$ over
$k$ with a Zariski dense set of $k$-rational points, and a regular function $f$
on $X$ inducing an injective map $X(k)\to k$ on $k$-rational points. In fact,
given any elliptic curve $E$ of positive rank over $k$, we can take $X=V\times
V$ with $V$ a suitable affine open set of $E$. The method of proof combines
value distribution theory for complex holomorphic maps with results of Faltings
on rational points in sub-varieties of abelian varieties.
