What is the arithmetic fundamental group? Let $X$ be an irreducible variety defined over $\mathbf{F}_p$. What's the "arithmetic fundamental group" of $X$? How does this relate to the algebraic fundamental group of a scheme? What's a good reference for this stuff?
 A: If $X$ is a connected scheme, then the arithmetic fundamental group of $X$ is the inverse limit of the automorphism groups $Aut(Y/X)$, where $Y\rightarrow X$ is finite etale and Galois.
It is also called the etale fundamental group of $X$ (sometimes the "etale" is omitted).
The reason for using the term "arithmetic" comes from considering the fundamental groups of varieties over, say, $\mathbb{Q}$. If $X$ is a variety over $\mathbb{Q}$, then by base change you can think of $X_\mathbb{C}$ as a variety over $\mathbb{C}$, which is a complex manifold, and thus has a topological fundamental group $\pi_1^{top}(X_{\mathbb{C}})$. By various comparison theorems (ie, "GAGA"), the etale fundamental group $\pi_1(X_{\mathbb{C}})$ is isomorphic to the profinite completion of $\pi_1^{top}(X_{\mathbb{C}})$.
On the other hand, the etale fundamental group of $X$ (ie, arithmetic fundamental group) is much larger than $\pi_1(X_{\mathbb{C}})$. This is because there are finite etale covers of $X$ which don't come from topological covers of the complex manifold $X_\mathbb{C}$. For example, if $K/\mathbb{Q}$ is a finite extension, then $X_K\rightarrow X$ is finite etale. By some other basic results we have $\pi_1(X_{\mathbb{C}}) = \pi_1(X_{\overline{\mathbb{Q}}})$, and in fact we have an exact sequence:
$$1\rightarrow \pi_1(X_{\overline{\mathbb{Q}}})\rightarrow \pi_1(X_{\mathbb{Q}})\rightarrow\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow 1$$
where the first term is called the geometric fundamental group (since it is isomorphic to the profinite completion of $\pi_1^{top}(X_{\mathbb{C}})$. The middle term thus combines this geometric part with the absolute Galois group of $\mathbb{Q}$ (the arithmetic part), which is why it is sometimes called the arithmetic fundamental group of $X$.
Returning to your question, by analogy with the case of schemes over $\mathbb{Q}$, in general if $X$ is a connected scheme over a field $K$, the geometric fundamental group of $X$ is $\pi_1(X_{K^{sep}})$, whereas the arithmetic fundamental group is just $\pi_1(X)$.
If you're interested in understanding the constructions and definitions in detail, Murre has some nice notes: (this is where I first learned about this stuff)
http://www.math.tifr.res.in/~publ/ln/tifr40.pdf
Lenstra also has some notes here:
http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf
I suppose the most canonical reference would be SGA 1 Expose V.
Though really there are a ton of references on this sort of stuff.
