Lehto-Virtanen-type and big Picard-type theorems for Berkovich analytic spaces (Algebraic Number Theory and Related Topics 2018)

Abstract

In non-archimedean setting, we establish a Lehto–Virtanen-type theorem for a morphism from the punctured Berkovich closed unit disk D {0} in the Berkovich affine line to the Berkovich projective line P1 having an isolated essential singularity at the origin, and then establish a big Picard-type theorem for such an open subset Ω in the Berkovich projective space P^[N] of any dimension N that the family of all morphisms from D {0} to Ω is normal in a non-archimedean Montel’s sense. As an application of the latter theorem, we see a big Brody-type hyperbolicity of the Berkovich harmonic Fatou set of an endomorphism of P^[N] of degree > 1.