What is the difference between $\pi_1 \mathbb{Z}_p$ and $\pi_1 \mathbb{Q}_p$?

Question

What is the difference between $\pi_1 \mathbb{Z}_p$ and $\pi_1 \mathbb{Q}_p$ (these are the etale fundamental group of $\mathrm{Spec}$ of the p-adic rings). I think (but I'm not sure) there is a map $$\pi_1\mathbb{Q}_p \to \pi_1\mathbb{Z}_p $$ given by sending an etale map $X \to Spec \mathbb{Q}_p$ $$X \to Spec \mathbb{Q}_p \to Spec \mathbb{Z}_p$$ My question is what are the fibers of this map, and is it surjective. And if not, what is a concrete example of something not in the image?

Answer 1

Recall that $\pi_1 \mathbb{Q}_p$ classifies connected finite etale coverings $X \to \operatorname{Spec} \mathbb{Q}_p$, which are all isomorphic to $\operatorname{Spec} F \to \operatorname{Spec} \mathbb{Q}_P$ for finite extensions $F/\mathbb{Q}_p$. Note that even though the map $\operatorname{Spec} \mathbb{Q}_p \to \operatorname{Spec} \mathbb{Z}_p$ is etale, it is not finite (nor is it a covering). So your map is not well defined.

On the other hand, the finite connected etale coverings $X \to \operatorname{Spec} \mathbb{Z}_p$ are all isomorphic to $\operatorname{Spec} \mathcal{O}_F \to \operatorname{Spec} \mathbb{Z}_p$, where $\mathcal{O}_F$ denotes the ring of integers in an unramified extension $F$ of $\mathbb{Q}_p$. Therefore, we can take such an $X \to \operatorname{Spec} \mathbb{Z}_p$ and consider the basechange \begin{align} X_{\mathbb{Q}_p} \to \operatorname{Spec} \mathbb{Q}_p, \end{align} which will be a finite etale covering. The construction $X \mapsto X_{\mathbb{Q_p}}$ defined a functor from the category of etale schemes $X \to \mathbb{Z}_p$ to the category of etale schemes over $\mathbb{Q}_p$, the induced map on etale fundamental groups is \begin{align} \pi_1 \mathbb{Q}_p \to \pi_1 \mathbb{Z}_p. \end{align}

[In general, given a morphism of schemes $X \to Y$, this procedure should give you a map $\pi_1 X \to \pi_1 Y $(up to choices of basepoints)]