## Anabelian puzzle 2: birational sections in the least non-abelian case

(Note:  this is being posted from an airport shortly before boarding, so this is less edited than usual.  I have some more concrete remarks on the puzzle below but will save them for later.)

Let X/K be a variety over a number field; then we have an exact sequence of etale fundamental groups

$\pi_1(X/\bar{K}) \rightarrow \pi_1(X/K) \rightarrow G_K$

and every point of X(K) induces a section from $G_K$ back to $\pi_1(X/K)$.  Grothendieck’s section conjecture asserts that, at least for certain classes of varieties X including projective smooth curves of genus greater than 1, the group-theoretic sections above are in fact in bijection with X(K).

A point P of X(K) is also a point of U(K), where U is any open subscheme of X containing P.  This gives you a section $G_K \rightarrow \pi_1(U/K)$ for any such U; the limit of all these fundamental groups is the Galois group of the maximal extension of the function field K(X) unramified at P.

In fact, you can associate to $P \in X(K)$ a section from G_K to the whole absolute Galois group of K(X) (or, better, a “bouquet” of related sections.)  The choice of P determines a decomposition group in G_{K(X)}, isomorphic to a semidirect product of its inertia group by a copy of G_K; now a section of this semidirect product is a section from G_K to G_{K(X)}.

So a weaker version of the section conjecture, the birational section conjecture, asserts that all of these sections from G_K to G_{K(X)} come from X(K).  Koenigsmann proved a few years ago that the birational section conjecture holds for K = Q_p.  A recent paper of Florian Pop proves something much stronger; that the birational section conjecture holds over finite extensions of Q_p even when G_{K(X)} is replaced with a really puny quotient.  Namely:  let K’ be the maximal elementary 2-abelian extension of Kbar(X), and let K” be the maximal elementary 2-abelian extension of K”.  Then you have a group Gamma described by an exact sequence

$1 \rightarrow G(K''/\bar{K}(X)) \rightarrow \Gamma \rightarrow G_K$

and what Pop proves is that this group Gamma, which you might call the “geometrically metabelian mod-2 fundamental group of X,” “remembers” enough about the curve X that already the sections from G_K to Gamma are all given by points of X.  Pop calls this a “minimalist birational section conjecture.”

One then wonders:  just how minimal can one get?  Abelianizing Gal(Kbar(X)) is too brutal; the resulting “geometrically abelian” fundamental group has lots of “extra” sections coming from points of Jac(X).  (Note:  just noticed this paper of Esnault and Wittenberg about exactly this, haven’t read it yet.)

Now here’s the puzzle — suppose we let K’ be the maximal elementary 2-abelian extension of Kbar(X) (i.e. the compositum of all quadratic extensions) and K” be the maximal elementary abelian 2-extension of K’ such that Gal(K”/Kbar(X)) has nilpotence class 2.  Then again you have

$1 \rightarrow G(K''/\bar{K}(X)) \rightarrow \Gamma' \rightarrow G_K$

where $\Gamma'$ could be called the “geometrically 2-nilpotent mod 2 fundamental group.”

So what are the sections from G_K to $\Gamma'$?  Is this fundamental group so minimalist that there are tons of extra sections, or is it the maximally minimalist context where a birational section conjecture could hold?

Update:  A bit jetlagged, but let me at least add one concrete question to this post.  I claim that a birational section as above would give you a function

$f: U(\mathbf{Q}) \rightarrow \mathbf{Q}^*/(\mathbf{Q}^*)^2$

for some open subscheme U of P^1, with the property that the Hilbert symbol

$(f(a)/(a-b), f(b)/(b-a))$

always vanishes.  Can you think of any such functions besides f(a) = a + constant, or f(a) = constant?