Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture

TL;DR

This paper proposes a new method to approach the geometric Bombieri-Lang conjecture for hyperbolic varieties, using entire curves constructed from rational points and relying on a non-degeneracy conjecture of partial heights.

Contribution

It introduces a novel approach linking rational points and entire curves, proving the conjecture for hyperbolic varieties with finite morphisms to abelian varieties.

Findings

Proves the geometric Bombieri-Lang conjecture for certain hyperbolic varieties.

Constructs entire curves from sequences of rational points.

Relies on a non-degeneracy conjecture of partial heights.

Abstract

We introduce a new approach to the geometric Bombieri--Lang conjecture for hyperbolic varieties in characteristic 0. The main idea is to construct an entire curve on a special fiber of a variety over a complex function field from an infinite sequence of rational points of the variety. The construction relies on the classical Brody lemma in complex geometry and is conditional on a non-degeneracy conjecture of partial heights. In particular, we prove the geometric Bombieri-Lang conjecture for hyperbolic projective varieties which have finite morphisms to abelian varieties over function fields of characteristic 0.