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.

中文翻译：

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二元多项式注入和椭圆曲线

对于每一个数字字段ķ，构造仿射代数表面X上ķ与Zariski密集的ķ -rational点，和常规的函数˚F上X诱导射地图\（X（k）的\ RIGHTARROWķ\）上ķ -理性点。实际上，给定任何超过k的正秩的椭圆曲线E，我们可以将\（X = V \ times V \）与V一起作为E的合适仿射开放集。证明方法将复杂全纯图的值分布理论与Abelian变体亚变种有理点的Faltings结果相结合。