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 ${\pi }_1^{\mathrm{et}}(X,x)$ (specifically: the algebraic fundamental group).

The term “anabelian” is alluding to “the less abelian ${\pi }_1^{\mathrm{et}}(X,x)$ 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 ${\pi }_1^{\mathrm{et}}(X,x)$ is called an anabelian variety.

An early conjecture motivating the theory (in Grothendieck) 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.

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

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.

There is some discussion of the area in

• Alexander Grothendieck, Esquisse d'un programme (1984).

A quick survey is on page 60 (2) of

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

A comprehensive introduction is in

• Fedor Bogomolov, Yuri Tschinkel, Introduction to birational anabelian geometry (pdf)

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