Anabelian geometry is a theory in number theory, which describes the way to which algebraic fundamental group G of a certain arithmetic variety V, or some related geometric object, can help to restore V. 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 H. Nakamura, A. Tamagawa, and complete proofs were given by Shinichi Mochizuki.
More recently, Sh. 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 algebrai fundamental group. Key results of mono-anabelian geometry were published in 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
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 Shinichi 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.
- http://www.math.jussieu.fr/~leila/SchnepsLM.pdf, p. 2.
- 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.
- http://www.math.sci.osaka-u.ac.jp/~nakamura/zoo/lion/INanabel.pdf, p. 2.
- Tamás Szamuely. "Heidelberg Lectures on Fundamental Groups" (PDF). section 5.