Strong Approximation

We first prove a technical lemma and corollary, then use them to deduce the strong approximation theorem, which is an extreme generalization of the Chinese Remainder Theorem; it asserts that is dense in the analogue of the adeles with one valuation removed.

The proof of Lemma 18.4.1 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.

Lemma 18.4.1   There is a constant that depends only on the global field with the following property:

Whenever is such that

 (18.8)

then there is a nonzero principal adele such that

for all $v$

Proof. This proof is modelled on Blichfeldt's proof of Minkowski's Theorem in the Geometry of Numbers, and works in quite general circumstances.

First we show that (18.4.1) implies that for almost all . Because is an adele, we have for almost all . If for infinitely many , then the product in (18.4.1) 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

will do. Thus suppose is as in (18.4.1).

The set of such that

 if is real archimedean if is complex archimedean if is non-archimedean

has measure

 (18.9)

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

Because of (18.4.2), in the quotient map there must be a pair of distinct points of that have the same image in , say

and

and

is nonzero. Then

for all . In the case of complex archimedean , we must be careful because the normalized valuation is the square of the usual archimedean complex valuation on , so e.g., it does not satisfy the triangle inequality. In particular, the quantity is at most the square of the maximum distance between two points in the disc in of radius , where by distance we mean the usual distance. This maximum distance in such a disc is at most , so is at most , as required. Thus satisfies the requirements of the lemma.

Corollary 18.4.2   Let be a normalized valuation and let be given for all with for almost all . Then there is a nonzero with

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

Proof. This is just a degenerate case of Lemma 18.4.1. Choose with and if . We can then choose so that

Then Lemma 18.4.1 does what is required.

Remark 18.4.3   The character group of the locally compact group is isomorphic to and plays a special role. See Chapter XV of [Cp86], Lang's [Lan64], Weil's [Wei82], and Godement's Bourbaki seminars 171 and 176. This duality lies behind the functional equation of and -functions. Iwasawa has shown [Iwa53] that the rings of adeles are characterized by certain general topologico-algebraic properties.

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

Theorem 18.4.4 (Strong Approximation)   Let be any normalized nontrivial valuation of the global field . Let be the restricted topological product of the with respect to the , where runs through all normalized valuations . Then  is dense in .

Proof. This proof was suggested by Prof. Kneser at the Cassels-Frohlich conference.

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

where is an open ball in about , and runs through finite sets of normalized valuations (not including ). Thus denseness of in is equivalent to the following statement about elements. Suppose we are given (i) a finite set of valuations , (ii) elements for all , and (iii) an . Then there is an element such that for all and for all with .

By the corollary to our proof that is compact (Corollary 18.3.6), there is a that is defined by inequalities of the form (where for almost all ) such that ever is of the form

 (18.10)

By Corollary 18.4.2, there is a nonzero such that

 for for

Hence on putting in (18.4.3) and multiplying by , we see that every is of the shape

where is the set of for . If now we let have components the given at , and (say) 0 elsewhere, then has the properties required.

Remark 18.4.5   The proof gives a quantitative form of the theorem (i.e., with a bound for ). For an alternative approach, see [Mah64].

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 2012-09-24