Greatest Common Divisors on the Complement of Numerically Parallel Divisors

2022/07/29 by Huang, Keping, Levin, Aaron · 2 citations
#11J87 (Primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.14432

Abstract

We prove inequalities involving greatest common divisors of functions at integral points with respect to numerically parallel divisors, generalizing a result of Wang and Yasufuku (after work of Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and the second author). After applying a result of Vojta on integral points on subvarieties of semiabelian varieties, we use geometry and the theory of heights to reduce to the (known) case of \mathbbG_mⁿ. In addition to proving results in a broader context than previously considered, we also study the exceptional set in this setting, for both the counting function and the proximity function.

Cited by

• Hyperbolicity and GCD for n+1 divisors with non-empty intersection
• Campana's orbifold conjecture for numerically equivalent divisors

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