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. A first version of Grothendieck anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves), then completed by Shinichi Mochizuki.[1]
^Shinichi Mochizuki, Hiroaka Nakamura, Akio Tamagawa, "The Grothendieck conjecture of the fundamental groups of algebraic curves", Sugaku Exposition (AMS English Translation) (14) 1, American Mathematical Society: 31-53, http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf (2001)
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Anabeliangeometry is a theory in number theory which describes the way in which the algebraic fundamental group G of a certain arithmetic variety X, or...
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ISBN 978-0-08-096019-7. MR 0930670. Stillwell, John (1994). Elements of Algebra: Geometry, Numbers, Equations. Springer-Verlag. pp. xi, 1–181. ISBN 9783540942900...
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