
Anabelian geometry

Anabelian geometry is a theory in number theory, which describes the way in which the algebraic fundamental group "G" of a certain arithmetic variety "V", or some related geometric object, can help to restore "V". 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) prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in "Esquisse d'un Programme" the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and , while complete proofs were given by Shinichi Mochizuki. 
More recently, Mochizuki introduced and developed a so called mono-anabelian geometry which restores, for a certain class of hyperbolic curves over number fields or some other fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry."
Anabelian geometry can be viewed as one of generalizations of class field theory. Unlike two other generalizations — abelian higher class field theory and representation theoretic Langlands program — anabelian geometry is highly non-linear and non-abelian. 
Formulation of a conjecture of Grothendieck on curves.
The "anabelian question" has been formulated as
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"). This was proved by Mochizuki. An example is for the case of formula_2 (the projective line) and formula_3, when the isomorphism class of "C" is determined by the cross-ratio in "K" of the four points removed (almost, there being an order to the four points in a cross-ratio, but not in the points removed). There are also results for the case of "K" a local field.