The non-Archimedean Green–Griffiths–Lang conjecture

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Non-archimedean Green–Griffiths–Lang conjecture

   Let $K$ be an algebraically closed complete non-archimedean valued field of characteristic zero, and let $X$ be a proper scheme over $K$. Suppose that, for every abelian variety $A$ over $K$, every morphism $A\to X$ is constant. Non-archimedean Green–Griffiths–Lang conjecture. Then $X$ is $K$-analytically Brody hyperbolic, meaning that for every finite type connected group scheme $G$ over $K$, every morphism $G^{\operatorname{an}}\to X^{\operatorname{an}}$ is constant. The conjecture is the non-archimedean analogue of the classical Green–Griffiths–Lang conjecture; the paper proves it in several cases but leaves the general statement open.

   source: Ariyan Javanpeykar and Alberto Vezzani, “Non-archimedean hyperbolicity and applications”, arXiv:1808.09880 (2021).