Bivariate polynomial injections and elliptic curves

Abstract

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 $$X(k)\rightarrow k$$ 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.