Anabelian geometry is a theory in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
which describes the way in which the
algebraic fundamental group
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a dat ...
''G'' of a certain
arithmetic variety ''X'', or some related geometric object, can help to restore ''X''. The first results for number fields and their absolute
Galois groups
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
were obtained by
Jürgen Neukirch,
Masatoshi Gündüz Ikeda,
Kenkichi Iwasawa
Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory.
Biography
Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gun ...
, and Kôji Uchida (
Neukirch–Uchida theorem
In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups.
showed that two algebraic number fields with the same absolute Galois group are isom ...
, 1969) prior to conjectures made about hyperbolic curves over number fields by
Alexander Grothendieck. As introduced in ''
Esquisse d'un Programme'' 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. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by
Shinichi Mochizuki.
Anabelian geometry can be viewed as one of the three generalizations of
class field theory. Unlike two other generalizations — abelian
higher class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is cred ...
and representation theoretic
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic n ...
— anabelian geometry is non-abelian and highly non-linear.
Formulation of a conjecture of Grothendieck on curves
The "anabelian question" has been formulated as
A concrete example is the case of curves, which may be
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substi ...
as well as projective. Suppose given a hyperbolic curve ''C'', i.e., the complement of ''n'' points in a projective
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
''g'', taken to be smooth and irreducible, defined over a field ''K'' that is finitely generated (over its
prime field
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
), such that
:
.
Grothendieck conjectured that the
algebraic fundamental group
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a dat ...
''G'' of ''C'', a
profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups ...
, 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
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
) and
, when the isomorphism class of ''C'' is determined by the
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
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
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...
.
Mono-anabelian geometry
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
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a dat ...
. 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.
Mono-anabelian geometry deals with certain types (strictly Belyi type) of hyperbolic curves over number fields and local fields. This theory considerably extends anabelian geometry. Its main aim to construct algorithms which produce the curve, up to an isomorphism, from the étale fundamental group of such a curve. In particular, for the first time this theory produces a simultaneous functorial restoration of the ground number field and its completion, from the fundamental group of a large class of punctured elliptic curves over number fields. Inter-universal Teichmüller theory of Shinichi Mochizuki is closely connected to and uses various results of mono-anabelian geometry.
Combinatorial anabelian geometry
Shinichi Mochizuki also introduced combinatorial anabelian geometry which deals with issues of hyperbolic curves and other related schemes over algebraically closed fields. The first results were published in Mochizuki's "A combinatorial version of the
Grothendieck conjecture" (2007) and "On the combinatorial cuspidalization of hyperbolic curves" (2010). The field was later applied to
hyperbolic curves by Yuichiro Hoshi and Mochizuki in a series of four papers, "Topics surrounding the combinatorial anabelian geometry of hyperbolic curves" (2012-2013).
Combinatorial anabelian geometry concerns the reconstruction of scheme- or ring-theoretic objects from more primitive combinatorial constituent data. The origin of combinatorial anabelian geometry is in some of such combinatorial ideas in Mochizuki's proofs of the Grothendieck conjecture. Some of the results of combinatorial anabelian geometry provide alternative proofs of partial cases of the Grothendieck conjecture without using p-adic Hodge theory. Combinatorial anabelian geometry helps to study various aspects of the Grothendieck-Teichmüller group and the absolute Galois groups of number fields and mixed-characteristic local fields.
See also
*
Class field theory
*
Fiber functor
*
Neukirch–Uchida theorem
In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups.
showed that two algebraic number fields with the same absolute Galois group are isom ...
*
Belyi's theorem
*
Frobenioid
*
Inter-universal Teichmüller theory
*
p-adic Teichmüller theory
*
Langlands correspondence
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic n ...
s
Notes
External links
*Foundations and Perspectives of Anabelian Geometry, RIMS workshop, June 28-July 2 2021. https://www.kurims.kyoto-u.ac.jp/~motizuki/RIMS-workshop-homepages-2016-2021/w1/May2020.html
*Combinatorial Anabelian Geometry and Related Topics, RIMS workshop, July 5-9 2021. https://www.kurims.kyoto-u.ac.jp/~motizuki/RIMS-workshop-homepages-2016-2021/w2/June2020.html
*
*
*
*
*The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves. http://www4.math.sci.osaka-u.ac.jp/~nakamura/zoo/rhino/NTM300.pdf
*Arithmetic fundamental groups and moduli of curves. http://users.ictp.it/~pub_off/lectures/lns001/Matsumoto/Matsumoto.pdf
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{{DEFAULTSORT:Anabelian Geometry
Arithmetic geometry