Adelic Algebraic Group - Ideles

An important example, the idele group I(K), is the case of . Here the set of ideles (correctly, idèles) consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles. Instead, considering that lies in two-dimensional affine space as the 'hyperbola' defined parametrically by

{(t, t−1)},

the topology correctly assigned to the idele group is that induced by inclusion in A2; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from A.

Inside AN, the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. In the case of the idele group, the quotient group

I(K)/K×

is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a compact group; the proof of this is essentially equivalent to the finiteness of the class number.

The study of the Galois cohomology of idele class groups is a central matter in class field theory. Characters of the idele class group, now usually called Hecke characters, give rise to the most basic class of L-functions.