# The Idele Group

The invertible elements of any commutative topological ring  are a group under multiplication. In general is not a topological group if it is endowed with the subset topology because inversion need not be continuous (only multiplication and addition on are required to be continuous). It is usual therefore to give the following topology. There is an injection

 (19.1)

of into the topological product . We give the corresponding subset topology. Then with this topology is a topological group and the inclusion map is continous. To see continuity of inclusion, note that this topology is finer (has at least as many open sets) than the subset topology induced by , since the projection maps are continuous.

Example 19.1.1   This is a non-example''. The inverse map on is continuous with respect to the -adic topology. If , then , so if , then

Definition 19.1.2 (Idele Group)   The idele group of is the group of invertible elements of the adele ring .

We shall usually speak of as a subset of , and will have to distinguish between the and -topologies.

Example 19.1.3   For a rational prime , let be the adele whose th component is and whose th component, for , is . Then as in , for the following reason. We must show that if is a basic open set that contains the adele , the for all sufficiently large are contained in . Since contains and is a basic open set, it is of the form

where if a finite set, and the , for , are arbitrary open subsets of that contain . If is a prime larger than any prime in , then for , is in . This proves convergence. If the inverse map were continuous on , then the sequence of would converge to . However, if is an open set as above about , then for sufficiently large , none of the adeles are contained in .

Lemma 19.1.4   The group of ideles is the restricted topological project of the with respect to the units , with the restricted product topology.

We omit the proof of Lemma 19.1.4, which is a matter of thinking carefully about the definitions. The main point is that inversion is continuous on for each . (See Example 19.1.1.)

We have seen that is naturally embedded in , so is naturally embedded in  .

Definition 19.1.5 (Principal Ideles)   We call , considered as a subgroup of , the principal ideles.

Lemma 19.1.6   The principal ideles are discrete as a subgroup of .

Proof. For is discrete in , so is embedded in by (19.1.1) as a discrete subset. (Alternatively, the subgroup topology on is finer than the topology coming from being a subset of , and is already discrete in .)

Definition 19.1.7 (Content of an Idele)   The content of is

Lemma 19.1.8   The map is a continuous homomorphism of the topological group into , where we view as a topological group under multiplication. If  is a number field, then is surjective.

Proof. That the content map  satisfies the axioms of a homomorphisms follows from the multiplicative nature of the defining formula for . For continuity, suppose is an open interval in . Suppose is such that . By considering small intervals about each non-unit component of , we find an open neighborhood of such that . It follows the is open.

For surjectivity, use that each archimedean valuation is surjective, and choose an idele that is  at all but one archimedean valuation.

Remark 19.1.9   Note also that the -topology is that appropriate to a group of operators on : a basis of open sets is the , where are, respectively, -compact and -open, and consists of the such that and .

Definition 19.1.10 (-Ideles)   The subgroup of -ideles is the subgroup of ideles such that . Thus is the kernel of , so we have an exact sequence

where the surjectivity on the right is only if is a number field.

Lemma 19.1.11   The subset of is closed as a subset, and the -subset topology on coincides with the -subset topology on .

Proof. Let with . To prove that is closed in , we find an -neighborhood of that does not meet .

1st Case. Suppose that (possibly ). Then there is a finite set  of  such that

1. contains all the with , and
2. .
Then the set can be defined by

for sufficiently small .

2nd Case. Suppose that . Then there is a finite set of such that

1. contains all the with , and
2. if an inequality implies (This is because for a non-archimedean valuation, the largest absolute value less than is , where is the residue characteristic. Also, the upper bound in Cassels's article is instead of , but I think he got it wrong.)
We can choose so small that (for ) implies Then may be defined by

This works because if , then either for all , in which case , so , or for some , in which case

so again .

We next show that the - and -topologies on are the same. If , we must show that every -neighborhood of contains an -neighborhood and vice-versa.

Let be an -neighborhood of . Then it contains an -neighborhood of the type

 (19.2) (19.3)

where is a finite set of valuations . This contains the -neighborhood in which in (19.1.2) is replaced by .

Next let be an -neighborhood. Then it contains an -neighborhood of the form

 (19.4) (19.5)

where the finite set  contains at least all archimedean valuations  and all valuations  with . Since , we may also suppose that is so small that (19.1.4) implies

Then the intersection of (19.1.4) with is the same as that of (19.1.2) with , i.e., (19.1.4) defines an -neighborhood.

By the product formula we have that . The following result is of vital importance in class field theory.

Theorem 19.1.12   The quotient with the quotient topology is compact.

Proof. After the preceeding lemma, it is enough to find an -compact set such that the map

is surjective. We take for the set of with

where is any idele of content greater than the of Lemma 18.4.1.

Let . Then the content of equals the content of , so by Lemma 18.4.1 there is an such that

all $v$

Then , as required.

Remark 19.1.13   The quotient is totally disconnected in the function field case. For the structure of its connected component in the number field case, see papers of Artin and Weil in the Proceedings of the Tokyo Symposium on Algebraic Number Theory, 1955'' (Science Council of Japan) or [AT90]. The determination of the character group of is global class field theory.

William Stein 2012-09-24