### 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 traditional conjectures, originating from Alexander Grothendieck and introduced in Esquisse d'un Programme were about how topological homomorphisms between two 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. Before anabelian geometry proper began with the famous letter to Gerd Faltings and Esquisse d'un Programme, the Neukirch–Uchida theorem hinted at the program from the perspective of Galois groups, which themselves can be shown to be étale fundamental groups.

More recently, Mochizuki introduced and developed a so called mono-anabelian geometry which restores, for a certain class of hyperbolic curves over number fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry."

Formulation of a conjecture of Grothendieck on curves

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

$$2-2g-n<0}.$$

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 $$g=0}$$ (the projective line) and n=4, 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.

Fiber functor
Neukirch–Uchida theorem
Belyi's theorem

Notes

Schneps, Leila (1997). "Grothendieck's "Long march through Galois theory"". In Schneps; Lochak, Pierre (eds.). Geometric Galois actions. 1. London Mathematical Society Lecture Note Series. 242. Cambridge: Cambridge University Press. pp. 59–66. MR 1483109.
Mochizuki, Shinichi (1996). "The profinite Grothendieck conjecture for closed hyperbolic curves over number fields". J. Math. Sci. Univ. Tokyo. 3 (3): 571–627. hdl:2261/1381. MR 1432110.
Ihara, Yasutaka; Nakamura, Hiroaki (1997). "Some illustrative examples for anabelian geometry in high dimensions" (PDF). In Schneps, Leila; Lochak, Pierre (eds.). Geometric Galois actions. 1. London Mathematical Society Lecture Note Series. 242. Cambridge: Cambridge University Press. pp. 127–138. MR 1483114.

Mochizuki, Shinichi (2003). "The absolute anabelian geometry of canonical curves" (PDF). Documenta Mathematica. Extra Vol., Kazuya Kato's fiftieth birthday: 609–640. MR 2046610.

Tamás Szamuely. "Heidelberg Lectures on Fundamental Groups" (PDF). section 5.
The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves. http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf
Arithmetic fundamental groups and moduli of curves. http://users.ictp.it/~pub_off/lectures/lns001/Matsumoto/Matsumoto.pdf

Alexander Grothendieck. "La Longue Marche à Travers la Théorie de Galois" (PDF).