The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck.
A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves), then completed by Shinichi Mochizuki.
[1] The "anabelian question" has been formulated as How much information about the isomorphism class of the variety X is contained in the knowledge of the étale fundamental group[2]A concrete example is the case of curves, which may be affine as well as projective.
Suppose given a hyperbolic curve C, i.e., the complement of n points in a projective algebraic curve of genus g, taken to be smooth and irreducible, defined over a field K that is finitely generated (over its prime field), such that Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e., the isomorphism class of G determines that of C).
[10] Shinichi Mochizuki also introduced combinatorial anabelian geometry which deals with issues of hyperbolic curves and other related schemes over algebraically closed fields.
Some of the results of combinatorial anabelian geometry provide alternative proofs of partial cases of the Grothendieck conjecture without using p-adic Hodge theory.