# absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$

In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{alg}$ are free profinite on countably many generators. However, they do not give reference, and after talking to some colleagues I am at a loss with regard to the status of this statement.

How does one derive this fact (if it is true)?

• I believe this is in the book "Field Arithmetic" by Fried and Jarden. Nov 20, 2017 at 18:01
• Actually, I could not find this in "Field Arithmetic." The statement that the Abelianization of the Galois group is a free, Abelian profinite group on countably many generators follows from Theorem 2.6 of the following article of Jarden and Pop, "Function Fields of One Variable over PAC Fields", Doc. Math., 14, pp. 517 -- 524: math.uni-bielefeld.de/documenta/vol-14/18.pdf Nov 20, 2017 at 18:33

The theorem is true. It seems to have been proved independently by Florian Pop and by David Harbater. In Pop's paper, it is the corollary on p. 556.

MR1334484 (96k:14011)
Pop, Florian
Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture.
Invent. Math. 120 (1995), no. 3, 555–578.