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

2023/05/24 by Junyi Xie, Xinyi Yuan, Xie, Junyi +1 · 3 citations
Mathematics · #11G35 #14G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2305.14789

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.

Cited by

• A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties
• Hilbert irreducibility for abelian varieties over function fields of characteristic zero
• Low degree subvarieties of universal hypersurfaces

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