Anabelian geometry is a proposed theory in mathematics, describing the way the algebraic fundamental group G of an algebraic variety V, or some related geometric object, determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian group, in a sense to be made more precise. The word anabelian (an alpha privative an- before abelian) was introduced in Esquisse d'un Programme, an influential manuscript of Alexander Grothendieck, circulated in the 1980s.
While the work of Grothendieck was for many years unpublished, and unavailable through the traditional formal scholarly channels, the formulation and predictions of the proposed theory received much attention, and some alterations, at the hands of a number of mathematicians. Those who have researched in this area have obtained some expected and related results, and in the 21st century the beginnings of such a theory started to be available.
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 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.
- Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), published in "Geometric Galois Actions", L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp. 5–48; English transl., ibid., pp. 243–283.
- 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.
- Heidelberg Lectures on Fundamental Groups, section 5.