現代数学の系譜工学物理雑談古典ガロア理論も読む78

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む78 (1002ﾚｽ)
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264: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/10/27(日) 12:30:17.87 ID:EUeYkluT

>>261

つづき

Generalizations of class field theory
There are three main generalizations, each of great interest on its own. They are: the Langlands program, anabelian geometry, and higher class field theory.

Often, the Langlands correspondence is viewed as a nonabelian class field theory. If/when fully established, it would contain a certain theory of nonabelian Galois extensions of global fields.
However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case.
It also does not include an analog of the existence theorem in class field theory, i.e. the concept of class fields is absent in the Langlands correspondence.
There are several other nonabelian theories, local and global, which provide alternative to the Langlands correspondence point of view.

Another generalization of class field theory is anabelian geometry which studies algorithms to restore the original object (e.g. a number field or a hyperbolic curve over it) from the knowledge of its full absolute Galois group of algebraic fundamental group.[3]

Another natural generalization is higher class field theory. It describes abelian extensions of higher local fields and higher global fields.
The latter come as function fields of schemes of finite type over integers and their appropriate localization and completions.
The theory is referred to as higher local class field theory and higher global class field theory. It uses algebraic K-theory and appropriate Milnor K-groups replace K_{1}}K_{1} which is in use in one-dimensional class field theory.
(引用終り)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/264

265: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/10/27(日) 12:30:44.74 ID:EUeYkluT

>>264

つづき

”The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions).
　Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups.[5]”
だとか

https://en.wikipedia.org/wiki/%C3%89tale_fundamental_group
Etale fundamental group
(抜粋)
The etale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Contents
1 Topological analogue/informal discussion
2 Formal definition
3 Examples and theorems
3.1 Schemes over a field of characteristic zero
3.2 Schemes over a field of positive characteristic and the tame fundamental group
3.3 Further topics
Further topics
From a category-theoretic point of view, the fundamental group is a functor

{Pointed algebraic varieties} → {Profinite groups}.
The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions).
Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups.[5]

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/265

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