Topics in absolute anabelian geometry. III: Global reconstruction algorithms (Q2797860)

The paper under review is the third part of a series (for part I, II, see [\textit{S. Mochizuki}, ibid. 19, No. 2, 139--242 (2012; Zbl 1267.14039); ibid. 20, No. 2, 171--269 (2013; Zbl 1367.14011)]). It is not easy to read because much of it consists of remarks, and there are many definitions which introduce new terminology. Its general topic are attempts to recover a scheme from its (profinite) fundamental group. The theorem of Neukirch-Uchida states that two number fields are isomorphic if their absolute Galois-groups are. The proof is indirect and shows that they cannot be different. However, for the fundamental group of a hyperbolic curve over a number field (which is an extension of the absolute Galois-group of the number field by the geometric fundamental group) the author shows how to recover more information about the field. The proof uses fully faithfulness of the fundamental group functor as well as Belyi's theorem.