Adele – c’est un nom si belle. (Oops, that’s bad French, isn’t it?)
I actually will not be able to finish the proof of the unit theorem here, because I don’t get to the ideles in this post. That will come next time (there are some of the same themes as here).
Let be a global field, i.e. a finite extension of either or . Then we can consider the set absolute values on . In the number field case, these are extensions (up to a power) of the archimedean absolute value on or the -adic absolute values by a theorem of Ostrowski classifying absolute values on . In the function field case, we need another result.
Here’s how we define the adele ring. It is the restricted direct product
where restricted means that any vector is required to satisfy for almost all . This becomes a topological ring if we take a basis of the form
where are open and is the ring of integers, and is a finite set containing the archimedean places. It is clear that addition and multiplication are continuous, and that is locally compact. For finite and containing the archimedean absolute values , there is a subring , and is the union of these subrings.
Since any is contained in for almost all (this is analogous to a rational function on a curve having only finitely many poles), there is an injective homomorphism .
Next, we may define a Haar measure on by taking the product of the Haar measures on , normalized such that for . Thus one gets a (i.e., the) Haar measure on itself. (more…)