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arXiv · 1811.02416

Bivariate polynomial injections and elliptic curves

Hector Pasten
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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 $X(k)\to k$ 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.

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Hector Pasten. 2018-11-06. Bivariate polynomial injections and elliptic curves. https://arxiv.org/abs/1811.02416

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