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 X, or some related geometric object, can help to restore X. 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. 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 Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki. Anabelian geometry can be viewed as one of the three generalizations of class field theory. Unlike two other generalizations — abelian higher class field theory and representation theoretic Langlands program — anabelian geometry is non-abelian and highly non-linear. 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 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 (the projective line) and , 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).