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Anabelian geometry
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Anabelian geometry
Anabelian geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related geometric object, can help to recover X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in Letter to Faltings (1983; see also Esquisse d'un Programme in 1984) the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves), then completed by Shinichi Mochizuki.
The theory has since grown in varieties (absolute, mono-anabelian, and combinatorial versions) and with multiple interactions with number theory, algebraic geometry, and low-dimensional topology.
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 Mochizuki. An example is for the case of (the projective line) and , 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.
Shinichi Mochizuki introduced and developed the mono-anabelian geometry, an approach which restores, for a certain class of hyperbolic curves over number fields or some other fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry" I (2012), II (2013), and III (2015).
The opposite approach of mono-anabelian geometry is bi-anabelian geometry, a term coined by Mochizuki in "Topics in Absolute Anabelian Geometry III" to indicate the classical approach.
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Anabelian geometry
Anabelian geometry is a theory in arithmetic geometry which describes the way in which the algebraic fundamental group of a certain arithmetic variety X, or some related geometric object, can help to recover X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in Letter to Faltings (1983; see also Esquisse d'un Programme in 1984) the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves), then completed by Shinichi Mochizuki.
The theory has since grown in varieties (absolute, mono-anabelian, and combinatorial versions) and with multiple interactions with number theory, algebraic geometry, and low-dimensional topology.
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 Mochizuki. An example is for the case of (the projective line) and , 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.
Shinichi Mochizuki introduced and developed the mono-anabelian geometry, an approach which restores, for a certain class of hyperbolic curves over number fields or some other fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry" I (2012), II (2013), and III (2015).
The opposite approach of mono-anabelian geometry is bi-anabelian geometry, a term coined by Mochizuki in "Topics in Absolute Anabelian Geometry III" to indicate the classical approach.