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

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.

The paper has been withdrawn by the author due to an error in the proof of Lemma 3.2. Section 1 up to Lemma 3.2 and Section 5 are to the best of the author's knowledge, still correct