In characteristic ( p > 0 ), the étale fundamental group of a smooth proper curve ( C ) of genus ( g \geq 2 ) is not known explicitly for any specific curve. While the prime-to-( p ) quotient ( \pi_1(C)^{(p)} ) is well-understood—being the profinite completion of the free group on ( 2g ) generators modulo the product of commutators ( w = \prod_{i=1}^g [x_i, y_i] )—the full étale fund...

Short answer: No. For a smooth proper curve C/k of genus g ≥ 2 over an algebraically closed field of characteristic p > 0, there is currently no nontrivial example for which the full geometric étale fundamental group π1(C) has been determined as an abstract profinite group.

What is known

• Prime-to-p part: For every ℓ ≠ p, the maximal pro-ℓ quotient is the pro-ℓ completion of the topological surf...

It is known that π₁(C) is topologically generated by 2g elements, meaning there is a continuous surjection from the profinite completion of the free group on 2g generato...

No – at the moment not a single smooth, proper, geometrically connected curve of genus ≥ 2 over an algebraically closed field of characteristic p > 0 is known for which the whole étale fundamental group has been determined.

What is known.

1. Genus 0: π₁(ℙ¹)=1 (the only finite étale covers of ℙ¹ are the trivial ones by the Riemann–Hurwitz formula).

2. Genus 1: For any abelian variety A (in...

Aside from the genus 0 and genus 1 cases, one really does ​not​ know any non‐trivial proper curve of genus ≥2 in char p whose ​full​ étale fundamental group is written down explicitly. In fact what one does know is:

1. By Grothendieck (SGA 1) the prime‐to‐p quotient π₁(C)^{(p′)}
   is exactly what you expect from characteristic 0, namely the profinite completion of the standard genus g ...

This is an excellent question that gets to the heart of some deep results in anabelian geometry and the arithmetic of curves in positive characteristic.

The short answer is: **Yes, the étale fundamental group is known for supersingular curves. For all other curves, its structure is known "in principle" but depends on a single, highly non-trivial relation that is generally not explicitly computabl...

The étale fundamental group of curves in characteristic is indeed a subtle and difficult topic. While we don't have complete descriptions for most curves of genus ≥ 2, there are some partial results and special cases where we know more.

Known Results and Special Cases

1. Structure of the Fundamental Group

The étale fundamental group sits in a short exact sequence:

Étale Fundamental Group of a Curve in Characteristic p

The étale fundamental group of a curve in positive characteristic is significantly more complex than in characteristic 0, primarily due to wild ramification. While a complete description for arbitrary curves of genus ≥ 2 remains challenging, there are some specific cases where substantial information is known:

Known Cases for Genus ≥ 2 ...

Let k be an algebraically closed field of characteristic p>0 and let
π₁(C)=π₁^{ét}(C) be the étale fundamental group of a smooth, proper, geometrically connected curve C/k of genus g.
With “known’’ we understand “described by an explicit presentation (generators and relations) as an abstract profinite group’’.

What is known in every characteristic

(1) The prime–to–p quotient
π₁(C)...