Adele ring

In mathematics, the adele ring is a construction in number theory that combines all local versions of a global field into one object. For the rational numbers, these local versions include the real numbers and the fields of ${\displaystyle p}$-adic numbers for all prime numbers ${\displaystyle p}$. More generally, if ${\displaystyle K}$ is a global field, its adele ring, often denoted ${\displaystyle \mathbb {A} _{K}}$, is a topological ring built from the completions ${\displaystyle K_{v}}$ of ${\displaystyle K}$ at all its places ${\displaystyle v}$. Formally, it is a restricted product of the local fields ${\displaystyle K_{v}}$, with respect to the valuation rings at the non-archimedean places. Its elements are called adeles.

The restricted product topology makes ${\displaystyle \mathbb {A} _{K}}$ a locally compact topological ring. The field ${\displaystyle K}$ embeds diagonally in ${\displaystyle \mathbb {A} _{K}}$ as a discrete subring, and the quotient ${\displaystyle \mathbb {A} _{K}/K}$ is compact. As an additive locally compact abelian group, the adele ring is self-dual, making it a natural setting for Fourier analysis on global fields.

The group of units of the adele ring, with its natural topology, is the idele group ${\displaystyle \mathbb {A} _{K}^{\times }}$. The quotient ${\displaystyle \mathbb {A} _{K}^{\times }/K^{\times }}$, called the idele class group, is a central object in class field theory. Adeles and ideles are also used in Tate's thesis, the theory of automorphic forms, local-global principles, and adelic descriptions of divisors, line bundles, and principal bundles on algebraic curves.