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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)?

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  • $\begingroup$ I believe this is in the book "Field Arithmetic" by Fried and Jarden. $\endgroup$
    – Jason Starr
    Commented Nov 20, 2017 at 18:01
  • $\begingroup$ 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 $\endgroup$
    – Jason Starr
    Commented Nov 20, 2017 at 18:33

1 Answer 1

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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.
http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002112868

Harbater's paper is the following.

MR1352282 (97b:14035)
Harbater, David
Fundamental groups and embedding problems in characteristic p. (English summary)
Recent developments in the inverse Galois problem (Seattle, WA, 1993), 353–369,
Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995.

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