Anabelian geometry

Not to be confused with Noncommutative geometry.

Anabelian geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related geometric object, can help to recover 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 Letter to Faltings (1983; see also Esquisse d'un Programme in 1984) 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. A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves), then completed by Shinichi Mochizuki.

The theory has since grown in varieties (absolute, mono-anabelian, and combinatorial versions) and with multiple interactions with number theory, algebraic geometry, and low-dimensional topology.

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