# 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?

• For a geometrically connected scheme $X$ of finite type over a field $k$, it is convenient to have distinct names for $\pi_1(X)$ and $\pi_1(X_{k_s})$ (linked through the exact sequence $1 \rightarrow \pi_1(X_{k_s}) \rightarrow \pi_1(X) \rightarrow {\rm{Gal}}(k_s/k) \rightarrow 1$). The latter is called the "geometric fundamental group" (since it coincides with $\pi_1(X_{\overline{k}})$), and the former is called the "arithmetic fundamental group". It would be entirely logical to drop the word "arithmetic", and it is fine to do so. But the convention persists, for emphasis I suppose. – nfdc23 Aug 17 '16 at 0:32

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.