nLab anabelian geometry

Idea

In anabelian geometry one studies how much information about a space $X$ (specifically: an algebraic variety) is contained already in its first étale homotopy group $\pi^{et}_1(X,x)$ (specifically: the algebraic fundamental group).

The term “anabelian” is supposed to be alluding to the fact that “the less abelian $\pi^{et}_1(X,x)$ is, the more information it carries about $X$.” Precisely: an anabelian group is a non-trivial group for which every finite index subgroup has trivial center.

Accordingly, an algebraic variety whose isomorphism class is entirely determined by $\pi^{et}_1(X,x)$ is called an anabelian variety.

An early conjecture motivating the theory (in Grothendieck 84) was that all hyperbolic curves over number fields are anabelian varieties. This was eventually proven by various authors in various cases. In (Uchida) and (Neukirch) it was shown that an isomorphism between Galois groups of number fields implies the existence of an isomorphism between those number fields. For algebraic curves over finite fields, over number fields and over p-adic field the statement was eventually completed by (Mochizuki 96).

Grothendieck also conjectured the existence of higher-dimensional anabelian varieties, but these are still very mysterious.

Étale homotopy types

• Alexander Schmidt, Jakob Stix, Anabelian geometry with étale homotopy types, (pdf).

Related entries

• algebraic fundamental group also called the ‘geometric fundamental

  group’ by Grothendieck.

• child's drawing/ Dessins d’enfant.

• inter-universal Teichmüller theory

• profinite completion

• Grothendieck-Teichmüller group

• section conjecture

• arithmetic geometry

• anabelioid

References

The notion of anabelian geometry originates in

• Alexandre Grothendieck, letter to Faltings (June 27, 1983) (pdf),

written in response to Faltings‘ work on the Mordell Conjecture, and the note of the Long March:

• Alexandre Grothendieck, La Longue Marche, Notes (link).

There is some discussion of the area in

• Alexander Grothendieck, Esquisse d'un programme

  (1984).

A relation with the theory of motives is in

• Alexander Schmidt, Motivic aspects of Anabelian geometry, Advanced studies in pure mathematics 63, 2012. Galois-Teichmüller theory and Arithmetic geometry. pp 503 - 517 pdf

Surveys:

• Yuri Tschinkel, Introduction to anabelian geometry, talk at Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends, Oxford 2014 (slides pdf)

• Leila Schneps, page 60 (2) of Grothendieck’s “Long march through Galois theory” (pdf)

A comprehensive introduction is in

• Fedor Bogomolov, Yuri Tschinkel, Introduction to birational anabelian geometry [pdf, arXiv:1011.0883]

• Hoshi, Yuichiro?. Introduction to Mono-anabelian Geometry. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2021), pp. 5-44. doi : 10.5802/pmb.42. http://www.numdam.org/articles/10.5802/pmb.42/

Examples are discussed in

• Yasutaka Ihara, Hiroaki Nakamura, Some illustrative examples for anabelian geometry in high dimensions, in Leila Schneps, P. Lochak (eds) Geometric Galois Actions I, London Math. Soc. Lect. Note Series 242 (pdf)

The classification of anabelian varieties for number fields was shown in

• J. Neukirch, Kennzeichnung der $p$-adischen und der endlichen algebraischen Zahlkörper, Invent. Math. 6 (1969), p. 296–314.

• J. Neukirch, Über die absoluten Galoisgruppen algebraischer Zahlkörper, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris (1977)

• K. Uchida. Isomorphisms of Galois groups, J. Math. Soc. Japan 28 (1976), no. 4, 617–620.

• K. Uchida, Isomorphisms of Galois groups of algebraic function fields, Ann. Math. (2) 106 (1977), no. 3, p. 589–598.

and for algebraic curves in

• Shinichi Mochizuki, The profinite Grothendieck conjecture for hyperbolic curves over number fields, J. Math. Sci. Univ. Tokyo 3 (1996), 571–627.

• Shinichi Mochizuki, The absolute anabelian geometry of canonical curves (2002) (pdf)

See also

• Frans Oort, Lecture notes. Informal notes (not for publication) made available for the Lorentz Center workshop ‘Anabelian number theory and geometry’, December 3-5, 2001

• N. V. Durov, Топологические реализации алгебраических многообразий (Topological realizations of algebraic varieties), preprint POMI 13/2012 (in Russian) abstract, pdf.gz

• Taylor Duypuy, Anabelian geometry

• Jackson Morrow, Kummer classes and Anabelian geometry, notes from Super QVNTS: Kummer Classes and Anabelian Geometry 2017 (pdf)