The proof of Lemma 20.4.1 below will use in a crucial way the normalized Haar measure on and the induced measure on the compact quotient . Since I am not formally developing Haar measure on locally compact groups, and since I didn't explain induced measures on quotients well in the last chapter, hopefully the following discussion will help clarify what is going on.

The real numbers under addition is a locally compact topological group. Normalized Haar measure has the property that , where are real numbers and is the closed interval from to . The subset of is discrete, and the quotient is a compact topological group, which thus has a Haar measure. Let be the Haar measure on normalized so that the natural quotient preserves the measure, in the sense that if is a measurable set that maps injectively into , then . This determine and we have since is a measurable set that maps bijectively onto and has measure . The situation for the map is pretty much the same.

*Whenever
is such that
*

for all $v$

First we show that (20.4.1) implies that for almost all . Because is an adele, we have for almost all . If for infinitely many , then the product in (20.4.1) would have to be 0. (We prove this only when is a finite extension of .) Excluding archimedean valuations, this is because the normalized valuation , which if less than is necessarily . Any infinite product of numbers must be 0, whenever is a sequence of primes.

Let be the Haar measure of induced from normalized Haar measure on , and let be the Haar measure of the set of that satisfy

if is real archimedean | ||

if is complex archimedean | ||

if is non-archimedean |

(As we will see, any positive real number would suffice in the definition of above. For example, in Cassels's article he uses the mysterious . He also doesn't discuss the subtleties of the complex archimedean case separately.)

Then since is compact, and because the number of archimedean valuations is finite. We show that

The set of such that

if is real archimedean | ||

if is complex archimedean | ||

if is non-archimedean |

has measure

(Note: If there are complex valuations, then the some of the 's in the product must be squared.)

Because of (20.4.2), in the quotient map there must be a pair of distinct points of that have the same image in , say

and

and

(all $v&ne#neq;v_0$)

We proved before that is discrete in . If one valuation is removed, the situation is much different.

Recall that if then a basis of open sets about is the collection of products

By the corollary to our proof that is compact (Corollary 20.3.6), there is a that is defined by inequalities of the form (where for almost all ) such that ever is of the form

By Corollary 20.4.2, there is a nonzero such that

for | ||

for |

Hence on putting in (20.4.3) and multiplying by , we see that every is of the shape

In the next chapter we'll introduce the ideles . Finally, we'll relate ideles to ideals, and use everything so far to give a new interpretation of class groups and their finiteness.

William Stein 2004-05-06