The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties

TL;DR

This paper proves the geometric Bombieri-Lang conjecture for certain projective varieties related to abelian varieties over function fields, using a novel approach involving entire curves and Lie algebras.

Contribution

It introduces a new method to describe entire curves via Lie algebras, advancing understanding of the conjecture for ramified covers of abelian varieties.

Findings

Proof of the geometric Bombieri-Lang conjecture in the specified setting

Explicit description of entire curves using Lie algebra structures

New techniques for analyzing ramified covers of abelian varieties

Abstract

In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of constructing entire curves in the pre-sequel "Partial heights, entire curves, and the geometric Bombieri-Lang conjecture." A new ingredient is an explicit description of the entire curves in terms of Lie algebras of abelian varieties.