Non-archimedean hyperbolicity and applications

Inspired by the work of Cherry, we introduce and study a new notion Brody hyperbolicity for rigid analytic varieties over non-archimedean field $K$ characteristic zero. We use this to show following algebraic statement: if projective variety admits non-constant morphism from an abelian variety, then so does any specialization it. As application result, that moduli space is $K$-analytically hyperbolic in equal These two results are predicted Green-Griffiths-Lang conjecture on its natural analogues hyperbolicity. Finally, Scholze's uniformization theorem prove aforementioned satisfies analogue "Theorem Fixed Part" mixed characteristic.