In mathematics, geometric class field theory is an extension of

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 credit ...

to higher-dimensional geometrical objects: much the same way as class field theory describes the

abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal s ...

of the

Galois group 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 pol ...

of a

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...

, geometric class field theory describes the abelianized

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of higher dimensional schemes in terms of data related to

algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the al ...

References

* {{cite book , last=Schmidt , first=Alexander , editor-first1=Stéphane, editor1=Ballet, editor-first2=Marc, editor2=Perret, editor-first3=Alexey, editor3= Zaytsev, title=Algorithmic arithmetic, geometry, and coding theory, publisher=Amer. Math. Soc. , date=2015 , pages=301–306 , chapter=A survey on class field theory for varieties, isbn=978-1-4704-1461-0 Class field theory Algebraic geometry