"Cute" applications of the étale fundamental group

Question

When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the circle that induces the identity on the circle. From this lemma, you easily deduce the Brouwer fixed point Theorem for the circle.

This was (for me) one of this "WOOOOW" moments where you realize that abstract constructions and some seemingly innocuous functorial lemmas may yield striking results (especially as I knew a quite long and complicated proof of Brouwer Theorem in dimension 2 before taking this topology class).

I was wondering if there exist (+ reference if they do) similarly "cute" applications of the construction of the étale fundamental group in Algebraic Geometry. Of course "cute" is not well-defined and may vary for each one of us, but existence of fixed points for the Frobenius morphism would I find especially cute. Any other relatively elementary result related to algebraic geometry over fields of positive characteristic will be appreciated!

Edit : I am obviously curious of any application of the étale fundamental group endowed with the aforementioned "WOOOW feeling". However, I'd be really interested in examples I could explain to smart grad students who are taking a first (but relatively advanced) course in Algebraic Geometry.

Answer 1

Using the étale fundamental group one can construct an injective group homomorphism

$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F_2})$

which is canonical in the sense that there are no choices involved in its construction (once the algebraic closure is fixed), $\operatorname{Out}$ refers to the outer automorphism group and $\widehat{F_2}$ to the profinite completion of the free group on two letters.

As for your example of a "WOOOOW" moment, this statement no longer contains the étale fundamental group in its statement, even though it's vital for the construction.

The fact that the absolute Galois group of the rationals is canonically a subgroup of the outer automorphisms of a (profinite-) free group is completely non-obvious. (Try to prove it from scratch...)

One can try to determine the image of this map. This leads to the profinite Grothendieck-Teichmueller group, which sometimes is conjectured to be isomorphic to $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

The étale fundamental group enters because one considers étale coverings of the projective line minus three points (i.e. the affine line minus two points). Over an algebraically closed field of characteristic zero, this has the étale fundamental group $\widehat{F_2}$ (think of the loops around two of the removed points as its generators). But now study the projective line minus three points over the rationals instead of their algebraic closure, this makes the Galois group of the rationals enter.

The projective line minus three points enters because by a construction due to Belyi, every algebraic curve which can be defined over a finite extension of the rationals can be realized as an étale covering of the projective line minus three points (this is an if and only if). The idea is to modify a map to the projective line so that its ramification gets concentrated in only three points.

You will find the rest of the story for example in surveys by Leila Schneps on the Grothendieck-Teichmueller theory.

Answer 2

I think a fantastic application of the étale fundamental group is in extensions of the Chabauty-Coleman(-Kim) method. The original idea was that we could bound the rational points on a curve $C$ by embedding it into the Jacobian and under good enough conditions (on the rank of the Jacobian compared to it's dimension), this leads to a proof of finiteness.

Well, what happens if these conditions aren't met? The idea is that we can replace the Jacobian (which can be thought of as coming from the abelian quotient of the fundamental group) with more non abelian analogues. This goes under the name of Chabauty-Coleman-Kim and Jennifer Balakrishnan, for instance, has lots of recent work on this topic.

Edit (adding in more details): The idea behind the original proof is simple: Suppose $C$ is a curve of genus $\geq 2$, $K$ is a number field and we want to prove that $C(K)$ is finite (Mordell's conjecture). For simplicity of notation, let me assume $K = \mathbb Q$

Under the Abel Jacobi embedding, we can think of $C$ as living inside the Jacobian ( with $g$ it's dimension) $$C(\mathbb Q) \subset J(\mathbb Q_p) \cong \mathbb Z_p^g\times\text{finite group}$$ If we consider the p-adic closure of the Mordell-Weil group (of rank $r$), $\overline{J(\mathbb Q)} \subset J(\mathbb Q_p)$ we get a lattice of rank $\leq r$ inside $\mathbb Z_p^g$ and let us suppose that $r+1 \leq g$. On the other hand $C(\mathbb Q_p)$ is a one dimensional p-adic curve and we have the inclusion: $$C(\mathbb Q) \subset C(\mathbb Q_p) \cap \overline{J(\mathbb Q)}$$ and we expect this intersection to be finite because we have a one dimensional thing intersection something of codimension at least 1. This can be made rigorous and even gives an effective bound under the assumption that $r + 1 \leq g$.

That's reasonable straightforward at least in concept but unfortunately, the restriction that $r < g$ is pretty severe and often won't hold. The really cool idea is that we can reinterpret this method in terms of data intrinsic to the curve $C$, in particular in terms of it's étale fundamental group. This paper by David Corwin gives a much better/complete exposition but let me try my best nevertheless.

We can reinterpret the Mordel-Weil group $J(\mathbb Q)\otimes\mathbb Z_p$ as living inside the cohomology group $H^1(\mathbb Q, T_p)$ where $T_p$ is the p-adic Tate module (ie, the abelian quotient of the fundamental group) and we identify which subgroup it corresponds to in terms of data intrinsic to $C$.What we need to do next is rewrite the entire Chabauty-Coleman method in terms of data that is intrinsic to $K$ (without reference to an Abelian variety) and then we can replace $T_p = \pi^{et}_C/[\pi^{et}_C,\pi^{et}_C]$ by a larger quotient by going deeper in the central filtration and it gets technically a lot more complicated but I hope that gives a general flavour of the idea.

Answer 3

The inverse Galois problem asks whether every finite group appears as the Galois group of some Galois extension of $\mathbb{Q}$. A well-known variant is the regular inverse Galois problem:

Let $G$ be a finite group. There is necessarily a Galois extension $K/\mathbb{Q}(t)$, regular over $\mathbb{Q}$, whose Galois group is isomorphic to $G$? Such an extension is said to be regular if there is no subextension of the form $L(t)$, where $L/\mathbb{Q}$ is non-trivial.

It is true that a positive answer to this question implies a positive answer to the real inverse Galois problem. This problem is still open as of today, but we can give an affirmative answer under some conditions on $G$ by exploring the relations between the Galois group and the étale fundamental group of the projective line.

Surprisingly, this is not that hard! A nice discussion about this may be found in the book Galois Groups and Fundamental Groups by Tamás Szamuely.