Non-Archimedean entire curves in projective varieties dominating an elliptic curve

TL;DR

This paper proves a non-Archimedean version of the Green--Griffiths--Lang conjecture for certain projective surfaces, showing they are hyperbolic when dominated by an elliptic curve, using non-Archimedean analytic methods.

Contribution

It establishes non-Archimedean hyperbolicity for projective surfaces of irregularity one that admit a dominant morphism to an elliptic curve, advancing non-Archimedean geometry.

Findings

Proves non-Archimedean hyperbolicity for specific surfaces

Shows algebraic degeneracy of entire curves in these varieties

Advances understanding of non-Archimedean Green--Griffiths--Lang conjecture

Abstract

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero. We prove the non-Archimedean Green--Griffiths--Lang conjecture for projective surfaces of irregularity one. More precisely, we prove that if $X/K$ is a groupless, projective surface that admits a dominant morphism an elliptic curve, then $X$ is $K$-analytically Brody hyperbolic. The main ingredient in our proof is a theorem concerning the algebraic degeneracy of non-Archimedean entire curves in projective, pseudo-groupless varieties admitting a dominant morphism to an elliptic curve.