Idele groups and class field theory are powerful tools in algebraic number theory. They connect local and global properties of number fields, providing a unified framework for studying abelian extensions and ideal class groups.
These concepts build on adeles, extending their applications to field extensions and reciprocity laws. Ideles offer a refined view of number fields, enabling deeper insights into their arithmetic structure and relationships between different algebraic objects.
Idele groups from adele rings
Construction and properties of idele groups
- Define idele groups as invertible elements in the adele ring of a number field
- Construct idele groups by selecting units from each local component of the adele ring
- Equip idele group JK of number field K with topology making it a locally compact topological group
- Topology of JK refined beyond subspace topology inherited from adele ring
- Define idele class group CK as quotient of JK by image of K* under diagonal embedding
- Idele class group CK exhibits compactness, crucial for class field theory applications
Canonical homomorphisms and unit ideles
- Establish canonical surjective homomorphism from JK to ideal group IK of K
- Map each idele to corresponding fractional ideal
- Identify kernel of JK to IK homomorphism as group of unit ideles
- Unit ideles consist of ideles with unit components at all finite places
- Surjective nature of homomorphism connects idele structure to ideal structure
- Kernel structure provides insight into relationship between local and global properties
Idele groups and ideal class groups
Quotient realizations and homomorphisms
- Realize ideal class group of number field K as quotient of idele class group CK
- Establish canonical surjective homomorphism from CK to ideal class group ClK
- Connect kernel of CK to ClK homomorphism with group of principal ideals
- Deduce finiteness of ideal class group from compactness of CK and discreteness of K* image in JK
- Unify treatment of finite and infinite places of K through idele formulation
- Enable study of generalized ideal class groups with modulus, crucial for comprehensive class field theory
Applications to number theory
- Provide natural setting for studying L-functions and zeta functions of number fields
- Offer refined view of ideal class group through idele class group connection
- Facilitate exploration of arithmetic properties through idelic structure
- Enable investigation of class numbers and related invariants using idelic approach
- Support analysis of prime decomposition in number field extensions via idelic formulation
- Enhance understanding of Dirichlet unit theorem through idelic perspective
Class field theory with ideles
Main theorems and correspondences
- State Existence Theorem: open subgroups of finite index in CK correspond to unique abelian extensions L/K
- Assert Isomorphism Theorem: Galois group of maximal abelian extension isomorphic to profinite completion of CK
- Formulate Conductor-Discriminant Formula using ideles, relating conductor, discriminant, and local factors
- Establish one-to-one correspondence between finite abelian extensions of K and open subgroups of finite index in CK (Main Theorem)
- Describe decomposition and inertia groups of primes in abelian extensions using idelic formulation
- Generalize class field theory to global fields of positive characteristic, unifying number and function fields
Applications to field extensions
- Provide natural framework for studying Hilbert class field of number field K
- Facilitate exploration of ray class fields using idelic approach
- Enable precise description of norm groups for abelian extensions
- Support investigation of ramification in abelian extensions through idelic formulation
- Offer tools for analyzing splitting of primes in abelian extensions
- Enhance understanding of Galois groups of abelian extensions using idelic structure
Artin reciprocity law with ideles
Idelic formulation and isomorphisms
- Establish Artin reciprocity law isomorphism between CK modulo norm group of L/K and Gal(L/K)
- Define idelic Artin map as homomorphism from JK to Gal(L/K)
- Induce isomorphism in Artin reciprocity law through idelic Artin map
- Identify kernel of idelic Artin map for L/K as norms of ideles from L to K
- Unify local reciprocity laws at all places of K (finite and infinite) through idelic formulation
- Express compatibility of Artin reciprocity with restriction and inflation of Galois groups using ideles
Applications and special cases
- Provide powerful tool for studying prime splitting behavior in abelian extensions
- Interpret explicit reciprocity laws (power residue symbols) as special cases of idelic Artin reciprocity
- Enable computation of Frobenius elements in Galois groups using idelic formulation
- Facilitate study of reciprocity laws in higher local fields through idelic approach
- Support investigation of Langlands program connections using idelic Artin reciprocity
- Enhance understanding of class field theory's local-global principles via idelic formulation