ideles

Ideles (or the idele group) of a global field F are the restricted direct product of the multiplicative groups of its local fields F_v over all places v of F, with respect to the units of the local rings of integers at finite places. Concretely, an idele is a tuple (x_v)_v with x_v ∈ F_v^× for each v, such that x_v ∈ O_v^× for all but finitely many non-Archimedean places v. For archimedean places, all components are allowed. The idele group I_F is a topological group under the restricted product topology and is locally compact and abelian.

Ideles unify local multiplicative data across all places. The field F embeds diagonally into I_F via x ↦ (x)_v, and the quotient I_F/F^× is the idele class group C_F. The idele class group, together with the global reciprocity map of class field theory, plays a central role in describing abelian extensions of F. In particular, there is a canonical topological isomorphism between the idele class group and the Galois group of the maximal abelian extension F^ab/F, given by the reciprocity map.

Hecke characters, or idele class characters, are continuous characters χ: I_F → C^× that are trivial on F^×, i.e., they factor through the idele class group. Such characters yield Hecke L-functions, generalizing Dirichlet L-functions, and are a primary tool in modern number theory.

A typical structural feature is the product formula: for any x ∈ F^×, the product over all places v of |x|_v equals 1, which explains why F^× sits inside the kernel of the global norm on I_F. For a concrete example, I_Q is R^× × ∏_p Q_p^× with almost all p-components in Z_p^×.

Ideles provide a unifying framework for global and local arithmetic, facilitating formulations of class field theory, automorphic forms, and L-functions.