The Arithmetic of Quaternion Algebra

46 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
norm na : H ×
A → A× too.
Suppose that G ′ is a group with unit 1, and for every place v ∈ V we have
defined the homomorphism fv Gv → G′ such that fv(Cv) = 1 p.p.. We can then
define in G ′ the product
fA(x) = �
fv(xv), if x = (xv) ∈ GA.
v∈V
EXAMPLE. We can define the norm NA, the modulus || · ||A in H ×
A , and A×
too.
NOTATIONS. It is convenient to consider Gv as that which has been embedded
in GA and identifies canonically with �
w�=v 1w × Gv, where 1w is the unit of
Gw, w ∈ V . When GA is the adele group of an algebraic group defined over K,
the group GK then is the group of points in G with its value in K. For every
place v ∈ V we choose an inclusion of GK in Gv denoted by iv. For almost
every place iv(GK) ⊃ Cv, then the mapping �
v∈V iv defines an inclusion of GK
in GA. We put X = XK = H or K, and Yv = Ov or Rv,p.p..
Quasi-characters. Recall that a quasi-character of a locally compact group is
a continuous homomorphism of the group in C × . Let ψA be a quasi-character
of GA. By restricting to Gv it defines a quasi-character ψv of Gv. We have
naturally the relation
ψA = �
ψv(xv)if x = (xv) ∈ GA.
v∈V
For the convergence of the product in C × if and only if ψv(Cv) = 1, p.p.. In
fact, if this property is not satisfied, we then could find cv ∈ Cv such that
|ψv(cv)−1| > 1/2, p.p. and the product would not be convergent for the elements
x such that xv = cvp.p.. Thus we have proved the following theorem.
Lemma 3.1.2. The mapping ψA ↦→ (ψv) is an isomorphism of the group of
quasi-characters of GA and the group {(ψv)} such that ψv is the quasi-character
of Gv, and ψv(Cv) = 1, p.p.
We can apply the local results of last chapter to the quasi-characters of XA.
Let ψA = �
v∈V ψv be the product of the canonical local character (exercise
II.4.1); the product is well-defined because of ψv(Yv) = 1, p.p.. The above
lemma shows that every character of XA is of the form x ↦→ ψa(ax), where
a = (av) ∈ Xv, and av ∈ Ker(ψv), p.p.. Since Ker(ψv) = Yv, p.p., it follows
that a ∈ A. Therefore, XA is self-dual. Let us turn firstly to the case where
X = Q or Fp(T ) is a prime field, we shall verify that ψA is trivial on XK, and
the dual of XA/XK is XK, cf. Weil [1].
Proposition 3.1.3. XA is self-dual, and XK is the dual of XA/XK.
We are going now to give the principal theorems of adeles XA and X ×
A .
These theorems are still valid if X is a central simple algebra over K. The proof
in the special case treated by us gives a good idea of the proof in the general
case (Weil [1]).
Theorem 3.1.4. (Fundamental Theorem) Adeles.
1) XK is discrete in XA and XA/XK is compact.