Idele group (${\mathbb{I}}_K$)

The idele group is the restricted product ${\prod }_v^{\prime }{K}_v^{\times }$ with respect to ${\mathcal{O}}_v^{\times }$ at finite places.

An idele obtained by mapping the same nonzero element of a number field into every completion. Only finitely many local valuations are nonzero, so this diagonal tuple belongs to the idele group.

The diagonal multiplicative group of a number field is discrete in its idele group. Require local units at every finite place and an archimedean neighbourhood of one. A diagonal element $x$ meeting these restrictions has integral $x-1$. If $x\ne 1$, its nonzero integer field norm cannot have absolute value less than one, while sufficiently small archimedean differences would force exactly that.

The map from an idele to its finite-place fractional ideal is surjective, has kernel the product of the infinite multiplicative groups and finite-place unit groups, and sends diagonal elements to principal fractional ideals. It realizes the ideal class group as the corresponding quotient of the idele group.

The idelic modulus is the product of normalized local moduli: ordinary real modulus, squared complex modulus and $(N\mathfrak{p}{)}^{-{\mathrm{ord}}_{\mathfrak{p}}(x)}$ at finite places. This continuous homomorphism to ${\mathbb{R}}_{>0}$ has the diagonal multiplicative group in its kernel by the product formula. It differs from the idele norm, which maps between idele groups of field extensions.

The norm-one idele group is the kernel of the idelic modulus. Its quotient by the diagonal $K^{\times }$ is compact for a number field and surjects onto the ideal class group.

An idele is an element of the idele group: a tuple of nonzero elements of the completion of a valued field at all places, with finite-place components in the local unit groups except at finitely many places.

The idele topology is the restricted product topology with respect to ${\mathcal{O}}_v^{\times }$. It is also the subspace topology induced by $x\mapsto (x,x^{-1})$ from ${\mathbb{A}}_K\times {\mathbb{A}}_K$.

The idele class group is $C_K={\mathbb{I}}_K/K^{\times }$, where $K^{\times }$ is embedded diagonally.

The ideles congruent to one modulo $\mathfrak{m}$ have finite components in $1+{\mathfrak{p}}^{n_{\mathfrak{p}}}$ at primes in the finite modulus and positive components at selected real places. Their image in $C_K$ is the idelic congruence subgroup.

For a finite abelian extension $L/K$, the norm group is the image of the idele-class norm $C_L\to C_K$. Class field theory gives $C_K/N_{L/K}{C}_L\cong \mathrm{Gal}(L/K)$.

The $v$-component of the idele norm is ${\prod }_{w\mid v}{N}_{L_w/K_v}(x_w)$. It descends to an idèle class group norm $C_L\to C_K$.

For a finite Galois extension $L/K$ with group $G$, the kernel of the global reciprocity map $C_K\to G^{\mathrm{ab}}$ is $N_{L/K}{C}_L$. It follows that this norm group equals the norm group from the maximal abelian subextension $L^{[G,G]}/K$.