Big Picard Theorem for jet differentials and Non-archimedean Ax-Lindemann Theorem

TL;DR

This paper extends Picard theorems to non-archimedean settings using jet differentials, leading to new hyperbolicity results including a non-archimedean Ax-Lindemann theorem and pseudo-Borel hyperbolicity.

Contribution

It introduces a non-archimedean big Picard theorem via jet differentials and applies it to establish hyperbolicity results for abelian varieties and their subvarieties.

Findings

Proves a non-archimedean Ax-Lindemann theorem for degenerate abelian varieties.

Establishes pseudo-Borel hyperbolicity for subvarieties of general type.

Generalizes Cherry and Ru's previous results using jet differential techniques.

Abstract

By implementing jet differential techniques in non-archimedean geometry, we obtain a big Picard type extension theorem, which generalizes a previous result of Cherry and Ru. As applications, we establish two hyperbolicity-related results. Firstly, we prove a non-archimedean Ax-Lindemann theorem for totally degenerate abelian varieties. Secondly, we show the pseudo-Borel hyperbolicity for subvarieties of general type in abelian varieties.