The de Jong fundamental group of a non-trivial abelian variety is non-abelian

TL;DR

This paper proves that the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension of b{Q}_p is non-abelian, extending previous results and showing its dependence on the base field.

Contribution

It generalizes the non-abelian nature of the de Jong fundamental group from b{P}^1_C to all non-trivial abelian varieties over certain fields and explores its dependence on the base field.

Findings

The de Jong fundamental group of non-trivial abelian varieties is non-abelian.

The fundamental group depends on the base field C.

It is 'big' for varieties admitting a non-constant map to projective space.

Abstract

We show that the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension $C/\mathbb{Q}_p$ is non-abelian. Generalizing an argument for $\mathbb{P}^1_C$, we also show that the de Jong fundamental group of any connected rigid analytic variety over $C$ admitting a non-constant map to $\mathbb{P}^n_C$ (e.g., an abelian variety) depends on $C$ and is big.