A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

162 CHAPTER 20. GLOBAL FIELDS AND ADELES
As already remarked, A +
K is a locally compact group, so it has an invariant
Haar measure. In fact one choice of this Haar measure is the product of the Haar
measures on the Kv, in the sense of Definition 20.2.5.
Corollary 20.3.7. The quotient A +
K /K+ has finite measure in the quotient measure
induced by the Haar measure on A +
K .
Remark 20.3.8. This statement is independent of the particular choice of the multiplicative
constant in the Haar measure on A +
K . We do not here go into the question
of finding the measure A +
K /K+ in terms of our explicitly given Haar measure. (See
Tate’s thesis, [Cp86, Chapter XV].)
Proof. This can be reduced similarly to the case of Q or F(t) which is immediate,
e.g., the W defined above has measure 1 for our Haar measure.
Alternatively, finite measure follows from compactness. To see this, cover AK/K +
with the translates of U, where U is a nonempty open set with finite measure. The
existence of a finite subcover implies finite measure.
Remark 20.3.9. We give an alternative proof of the product formula |a| v = 1
for nonzero a ∈ K. We have seen that if xv ∈ Kv, then multiplication by xv
magnifies the Haar measure in K + v by a factor of |xv| v . Hence if x = {xv} ∈ AK,
then multiplication by x magnifies the Haar measure in A +
K by |xv| v . But now
multiplication by a ∈ K takes K + ⊂ A +
K into K+ , so gives a well-defined bijection
of A +
K /K+ onto A +
K /K+ which magnifies the measure by the factor
|a| v . Hence
|a|v = 1 Corollary 20.3.7. (The point is that if µ is the measure of A +
K /K+ , then
µ = |a| v · µ, so because µ is finite we must have |a| v = 1.)
20.4 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 K + is dense in the analogue of the adeles with one
valuation removed.
The proof of Lemma 20.4.1 below will use in a crucial way the normalized Haar
measure on AK and the induced measure on the compact quotient A +
K /K+ . 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 R + under addition is a locally compact topological group.
Normalized Haar measure µ has the property that µ([a, b]) = b − a, where a ≤ b
are real numbers and [a, b] is the closed interval from a to b. The subset Z + of R +
is discrete, and the quotient S1 = R + /Z + is a compact topological group, which
thus has a Haar measure. Let µ be the Haar measure on S1 normalized so that the
natural quotient π : R + → S1 preserves the measure, in the sense that if X ⊂ R +
is a measurable set that maps injectively into S1 , then µ(X) = µ(π(X)). This