Finiteness criteria and uniformity of integral sections in some families of abelian varieties

2019/12/06 by Phung, Xuan Kien
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.02930

Abstract

Let A be abelian variety over the function field K of a compact Riemann surface B. Fix a model f \colon A → B of A/K and a certain effective horizontal divisor \DD ⊂ A. We give a sufficient condition on the divisor \DD for the finiteness of the set of (S, \DD)-integral sections for every finite subset S ⊂ B. These integral sections σ correspond to rational points in A(K) which satisfy the geometric condition f ( σ(B) ∩ \DD)⊂ S. This notion is the geometric variant of integral solutions of a system of Diophantine equations. When A= A₀ × B for some complex abelian variety A₀, we also give a certain uniform bound on the number of (S, \DD)-integral sections. For trivial families of abelian surfaces, a numerical criterion on \DD for the finiteness of (S, \DD)-integral sections is obtained.

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