Integral points on varieties with infinite étale fundamental group

2022/12/12 by Niven Achenjang, Achenjang, Niven T., Jackson S. Morrow +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2212.06210

openalex publication_date 2022/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 φ\colon Y → X of degree at least 2dim(X)², there exists an ample line bundle \mathscrL on Y such that for a general member D of the complete linear system |\mathscrL|, D is geometrically irreducible and any set of φ(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.

Related

• Fundamental groups of proper varieties are finitely presented
• Existence of rational points as a homotopy limit problem
• Some fundamental groups in arithmetic geometry
• Finiteness of étale fundamental groups by reduction modulo p
• Arithmetic representations of fundamental groups II: finiteness