Generalized integral points on abelian varieties and the Geometric Lang-Vojta conjecture

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

paper · doi:10.48550/arxiv.1912.02932

Abstract

Let A be an abelian variety over the function field K of a compact Riemann surface B. Fix a model f \colon A → B of A/K and an effective horizontal divisor D ⊂ A. We study (S, D)-integral sections σ of A where S ⊂ B is arbitrary. These sections σ are algebraic and satisfy the geometric condition f(σ(B) ∩ D)⊂ S. Developing the idea of Parshin, we formulate a hyperbolic-homotopic height of such sections as a substitute for intersection theory to establish new results concerning the finiteness and the polynomial growth of large unions of (S, D)-integral points where S is only required to be finite in a thin analytic open subset of B. Such results are out of reach of purely algebraic methods and imply new evidence and interesting phenomena to the Geometric Lang-Vojta conjecture.

Related

• Finiteness criteria and uniformity of integral sections in some families of abelian varieties
• Some cases of Vojta's Conjecture on integral points over function fields
• Large unions of generalized integral sections on elliptic surfaces
• Splitting of almost ordinary abelian surfaces in families and the S-integrality conjectures
• Elliptic curves over function fields with a large set of integral points