A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties

2025/11/21 by Guoquan Gao, Gao, Guoquan
Mathematics · #11G35 #11J97 #14G05 #32H30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2511.17010

openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/30

Abstract

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the hyperbolicity assumption or the non-isotriviality assumption. The proof builds upon their strategy for constructing entire curves, yet hinges crucially on a new counting argument and draws substantially on tools from algebraic geometry and Nevanlinna theory to overcome various technical difficulties.

Citations

• Hilbert irreducibility for abelian varieties over function fields of characteristic zero
• Parshin's method and the geometric Bombieri-Lang conjecture
• Finiteness of pointed maps to moduli spaces of polarized varieties
• The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties
• A note related to Severi's finiteness theorem
• Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture
• On the distribution of rational points on ramified covers of abelian varieties
• Existence of minimal models for varieties of log general type
• Shokurov's Rational Connectedness Conjecture

Related

• The Geometric Bombieri-Lang Conjecture for Ramified Covers of Abelian Varieties
• Parshin's method and the geometric Bombieri-Lang conjecture
• Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture
• Rational points and ramified covers of products of two elliptic curves
• A higher dimensional Hilbert irreducibility theorem