Reconstruction of function fields (Q938226)

Let \(K=k(X)\) be the rational function field of a surface \(X\) over \(k\) an algebraic closure of the field with \(p\) elements. The authors show that \(K\) can be recovered from the maximal pro-\(\ell\)-quotient \(\mathcal G_K\) of its absolute Galois group; namely, if there exists an isomorphism \(\Psi\) from the abelianization \(\mathcal G_K ^a\) to \(\mathcal G_L ^a\) of abelian pro-\(\ell\)-groups including a bijection of sets \(\Sigma _K =\Sigma _L\) for another function field \(L/k\), then \(c\Psi\) is induced by an isomorphism between the perfect closures of \(K\) and \(L\), where \(\Sigma _K\) is determined by the second lower central series quotient of \(\mathcal G_K\), and \(c\in \mathbb{Z}^* _\ell\). This long paper has contents including projective structures, flag maps, valuations, Galois groups of curves, valuations on surfaces, \(\ell\)-adic analysis (curves, surfaces, etc), and ends with a 3-page proof.