Around the Grothendieck anabelian section conjecture (Q2900373)

``This, mostly expository, paper is built around the topic of the Grothendieck anabelian section conjecture. This conjecture predicts that splittings, or sections, of the exact sequence of the arithmetic fundamental group NEWLINE\[NEWLINE 1\to \pi_1(\bar X)\to \pi_1(X)\to G_k\to 1 NEWLINE\]NEWLINE of a proper, smooth, and hyperbolic curve \(X\), all arise from decomposition subgroups associated to rational points of \(X\)'' over a certain arithmetic base field \(k\). The author introduces a notion of \textit{(uniformly) good group-theoretical section}, that behaves nicely with local-global principle in \(k\), which can be lifted to sections into the \textit{cuspidally abelian quotient} of the absolute Galois group of the function field \(k(X)\). The main ingredients are based on a preprint whose expanded version has been published as [\textit{M. Saïdi}, Adv. Math. 230, No. 4--6, 1931--1954 (2012; Zbl 1260.14036); J. Pure Appl. Algebra 217, No. 3, 583--584 (2013; Zbl 1262.14029)].NEWLINENEWLINEFor the entire collection see [Zbl 1237.11001].