On a problem posed by Bjorn Poonen

Abstract

Bjorn Poonen asked whether there exists a polynomial giving a surjection $\mathbb{Z}\times \mathbb{Z}\to \mathbb{N}$. We answer this question in the negative, conditional on a conjecture of Vojta. More precisely, we show that if such a function exists, there is a family of open surfaces with dense integral points despite the surfaces being of log general type.

11 pages, 1 figure. Comments welcome!