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, Niven T Achenjang Department of Mathematics, Massachusetts Institute of Technology , Cambridge, MA 02139, USA Search for other works by this author on: Oxford Academic Jackson S Morrow Department of Mathematics, University of California, Berkeley , Berkeley, CA 94720, USA Correspondence to be sent to: e-mail: jacksonmorrow@berkeley.edu Search for other works by this author on: Oxford Academic
International Mathematics Research Notices, Volume 2024, Issue 10, May 2024, Pages 8157–8171, https://doi.org/10.1093/imrn/rnad147
Published:
12 July 2023
Article history
Received:
19 January 2023
Revision received:
26 May 2023
Accepted:
16 June 2023
Published:
12 July 2023
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Niven T Achenjang, Jackson S Morrow, Integral Points on Varieties With Infinite Étale Fundamental Group, International Mathematics Research Notices, Volume 2024, Issue 10, May 2024, Pages 8157–8171, https://doi.org/10.1093/imrn/rnad147
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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 |$\varphi \colon Y \to X$| of degree at least |$2\dim (X)^{2}$|, there exists an ample line bundle |$\mathscr{L}$| on |$Y$| such that for a general member |$D$| of the complete linear system |$|\mathscr{L}|$|, |$D$| is geometrically irreducible and any set of |$\varphi (D)$|-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.
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