Integral points on varieties with infinite étale fundamental group

Abstract

We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\phi :Y\to X$ of degree at least $2\dim (X{)}^2$, there exists an ample line bundle $\mathcal{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathcal{L}|$, $D$ is geometrically irreducible and any set of $\phi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.

arXiv commentv2: 10 pages. Fixed an error in previous version and strengthened results. Final version