From the authors’ introduction: “A major open problem today is to identify classes of fields characterized by their absolute Galois groups. There exist genuinely different fields with isomorphic Galois groups, e.g. \(\mathbb{F}_p\) and \(\mathbb{C}((t))\). However, J. Neukirch [Invent. Math. 6, 296–314 (1969; Zbl 0192.40102)] and K. Uchida [Ann. Math. (2) 106, 589–598 (1977; Zbl 0372.12017)] showed that Galois groups of maximal solvable extensions of number fields or function fields of curves over finite fields determine the corresponding field, up to isomorphism. This result is the first instance of birational an abelian geometry [which] aims to show that Galois groups of function fields of algebraic varieties over an algebraically closed ground field determine the function field in a functorial way. The version proposed by Grothendieck introduces a class of an abelian varieties, functorially characterized by their étale fundamental groups (…) However, absolute Galois groups are simply too large. It turns out that there are intermediate groups, whose description involves some projective geometry – most importantly, geometry of lines and points in the projective plane (…) These groups are just minimally different from abelian groups; they encode the geometry of simple configurations. On the other hand, their structure is already sufficiently rich so that the corresponding objects in the theory of fields allow us to capture all invariants and individual properties of large fields, i.e., function fields of transcendence degree at least two over algebraically closed ground fields. This insight of the first author was developed in the series of papers written at the Courant Institute over the last decade”. The authors’ article in current developments in algebraic geometry [Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 59, 17–63 (2012; Zbl 1290.14017)], gave a survey of the development of the main ideas merging into this program. Here the authors prove a few new results and “formulate their vision of the future directions of research”, which roughly belong to:
1. Projective geometry and \(K\)-theory:
Abstract (axiomatized) projective spaces are related to Galois theory by the following result: “If \(K/k\) is an extension of fields, then \(K^*/k^*\) is simultaneously an abelian group and a projective space. Conversely, an abelian group with a compatible projective structure corresponds to a field extension”. Over non-closed fields \(K\), projective spaces admit non-trivial forms, called Brauer-Severi varieties, which are classified by the Brauer group \(\text{Br}(K)= H^2(G_K, \mathbb{G}_m)\), which is in turn closely related to the Milnor group \(K^M_2(K)\). The authors show that if \(K\) and \(L\) are function fields of algebraic varieties of dimension \(\geq 2\) over algebraically closed fields \(k\) and \(l\), such that there exist compatible isomorphisms of abelian groups \(\psi_1: K_1(K)\to K_1(L)\), \(\psi_2: K_2(K)\to K_2(L)\), then there exists an isomorphism of fields \(\psi: K\to L\) such that the induced map on \(K^*\) coincides with \(\psi^{\pm 1}_1\). The main point is that \(K_2(K)\) encodes the canonical projective structure of \(K^*/k^*\).
2. Projective geometry and Galois groups:
Let \(G_K\) be the absolute Galois group of a field \(K\), \(G^a_K= G_K/[G_K, G_K]\). \(G^c_K= G_K/[G_K, [G_K, G_K]]\). Fix a prime \(\ell\neq p\) and denote by \({\mathcal G}_K\), \({\mathcal G}^a_K\) and \({\mathcal G}^c_K\), respectively the maximal pro-\(\ell\)-quotients of the previous Galois groups. Define the fan \(\Sigma_K\) on \({\mathcal G}_K\) to be the set of all topological subgroups \(\sigma\) of \({\mathcal G}^a_K\) such that the preimage of \(\sigma\) under the natural projection \({\mathcal G}^c_K\to{\mathcal G}^a_K\) is abelian. The authors show that if \(K\) and \(L\) are function fields of transcendence degree \(\geq 2\) over \(k=\overline{\mathbb{F}_p}\) and if \(\psi^*:{\mathcal G}^a_L\to{\mathcal G}^a_K\) is an isomorphism of abelian pro-\(\ell\)-groups inducing a bijection of sets \(\Sigma_L=\Sigma_K\), then there exists an isomorphism of perfect closures \(\overline\psi:\overline K\to\overline L\), unique modulo rescaling by a constant in \(\mathbb{Z}^*_{\ell}\). The main point is that the Galois group \({\mathcal G}^c_K\) encodes information about affine and projective structures on \({\mathcal G}^a_K\) in close parallel to what happened in the context of \(K\)-theory in Section 1.
3. \(\mathbb{Z}\)-version of the Galois group:
Here the authors introduce a functorial version of the reconstruction/recognition processes presented in the two previous sections. It will allow to recover not only field isomorphisms from \(K\)-theoretic or Galois-theoretic data, but also sections, i.e., rational points. More precisely, if \(K\) and \(L\) are function fields over algebraic closures of finite fields \(k\) and \(l\), respectively, suppose that: (a) \(\psi_1: K^*/k^*\to L^*/l^*\) is a homomorphism such that for any one-dimensional subfield \(E\) of \(K\), there exist a one-dimensional subfield \(F\) of \(L\) s.t. \(\psi_1(E^*/k^*)\subseteq F^*/l^*\); (b) \(\psi_1(K^*/k^*)\) contains at least two algebraically independent elements of \(L^*/l^*\). Then, if \(\psi_1\) is injective, there exists a subfield \(F\) of \(L\), a field isomorphism \(\phi: K\underset\widetilde{}\rightarrow F\) and an integer \(m\in \mathbb{Z}\) such that \(\psi_1\) coincides with the homomorphism induced by \(\phi^m\) (if \(\psi_1\) is not injective, there is a more complicated statement).
4. Galois cohomology:
By duality, the main result of Section 3 confirms the general concept that birational properties of algebraic varieties are functorially encoded in the structure of \(G^c_K\). On the other hand, the conjecture of Bloch-Kato (now a theorem of Voevodsky et al.) states that for any field \(K\) and any prime \(\ell\), the Galois symbol induces an isomorphism \(K^M_n(K)\underset\widetilde{}\rightarrow H^n(G_K,\mu^{\otimes n}_{\ell})\) for any \(n\in\mathbb{N}\). The authors show that, for \(k=\overline{\mathbb{F}_p}\), \(p\leq \ell\) and \(K= k(X)\), of dimension \(\geq 2\), the Bloch-Kato conjecture for \(K\) is equivalent to the surjectivity of the map \(\pi^*_a: H^*({\mathcal G}^a_K, \mathbb{Z}/\ell^n)\to H^*(G_K, \mathbb{Z}/\ell^n)\) induced by the projection \(\pi_a: G_K\to{\mathcal G}^a_K\), and the equality \(\ker(\pi^*_a)= \ker(\pi^*_c)\), where \(\pi^*_c\) is defined analogously to \(\pi^*_a\). This implies that the Galois cohomology of \({\mathcal G}_K\) encodes important birational information on \(X\). In the case above, \({\mathcal G}_K\), and hence \(K\), modulo purely inseparable extensions, can be recovered from the cup products \(H^1({\mathcal G}_K, \mathbb{Z}/\ell^n)\otimes H^1({\mathcal G}_K, \mathbb{Z}/\ell^n)\to H^2({\mathcal G}_K, \mathbb{Z}/\ell^n)\), \(n\in\mathbb{N}\) (NB: compare with the main results of Chebolu et al. [Math. Ann. 352, No. 1, 205–221 (2012; Zbl 1272.12015)] on the \(\ell^a\)-central descending series for a field containing a primitive \(\ell^a\)th root of unity).
5. Freeness:
For for \(k=\overline{\mathbb{F}_p}\), \(p\neq \ell\) and \(K= k(X)\), of dimension \(\geq 2\), assume moreover that \(X\) has dimension \(n\) and contains a smooth \(k\)-rational point. The first author has shown that, for any \(\ell\)-Sylow subgroup \({\mathfrak E}_{\ell}\) of \(G_K\), \({\mathfrak E}_{\ell}(G_K)={\mathfrak E}_{\ell}(G_{k(\mathbb{P}^n)})\); in particular, when \(k\) is algebraically closed, \({\mathfrak E}_l\) depends only on the dimension of \(X\). He conjectures that if \(k\) is algebraically closed, then \(H^i([{\mathfrak E}_{\ell},{\mathfrak E}_{\ell}],\mathbb{Z}/\ell^m)= 0\) for all \(i\geq 2\), \(m\in\mathbb{N}\). Note that this freeness conjecture implies the Bloch-Kato conjecture for \(K= k(X)\). Considerations of spectral sequences allow to complement it by the following conjecture: Let \({\mathfrak E}_{\ell}^{(j)}\) be the derived series of \({\mathfrak E}_{\ell}\) and \(M= \operatorname{Hom}({\mathfrak E}_{\ell}^{(1)}/{\mathfrak E}_{\ell}^{(2)}, \mathbb{Z}/\ell^m)\). Then the projection \(M\to M/M^{{\mathfrak E}^a_{\ell}}\) can be factored as \(M\rightarrowtail D\twoheadrightarrow M/M^{{\mathfrak E}^a_{\ell}}\), where \(D\) is a cohomologically trivial \({\mathfrak E}^a_{\ell}\)-module.