The Arithmetic of Quaternion Algebra

58 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
everywhere, hence ZL(s) = ZK(s) 2 has a double pole at s = 1, this implies that
L is not a field and then a ∈ K ×2 .
n = 3, q(x, y, z) = ax2 + by2 + z2 , up to equivalence on K. Choosing H to be
the quaternion algebra associating to {a, b}, the principle is equivalent to the
characterization of matrix algebra.
n ≥ 4, It turns back by induction to the precedent cases, cf. Lam [1],p.170.
Since JH and ZK satisfy the functional equations:
JH(s) = (−1) |Ram(H)|
�
p∈Ramf (H)
ZK(s) = D s−1/2
K ZK(s).
We obtain a functional equation for ZH:
Np 1−s · ZH(2 − s),
ZH(s) = (D 4 HN(dH) 2 ) 1
2 −s (−1) |Ram(H)| ZH(1 − s)
which, if comparing it with the functional equation (thm.2.2)ZH = D 1
2 −s
H ZH(1−
s) obtained directly when H is a field, shows that DH = D 4 K N(dH) 2 , but
immediately:
Property II.
The number of places ramified in a quaternion algebra is even.
In the case of characteristic different from 2, this statement is equivalent to the
reciprocal formula of Hilbert symbol.
Corollary 3.3.3. (Reciprocal formula of Hilbert symbol). Let K be a global
field of characteristic different from 2. For two elements a, b of K × , let (a, b)v
be their Hilbert symbol on Kv. We have the product formula
�
(a, b)v = 1
where the product takes on every place v of K.
v
Application:
1) Choosing K = Q and for a, b two odd prime numbers, one can verifies the
process for obtaining the quadratic reciprocal formula
2)Computation of symbol (a, b)2. The Hilbert symbol of two rational numbers
a, b on Qp can be computed easily with the rule described in II,§1. We shall
calculate (a, b)2 by using the product formula:(a, b)2 = �
v�=2
(a, b)v.
Before proving the property of existence of a quaternion algebra of the given
local Hasse invariants, we extract some consequences of property I and property
II above. The extension L/K are always assumed to be separable.
Corollary 3.3.4. (Norm theorem in quadratic extension). Let L/K be a separable
quadratic extension, and θ ∈ K × . For θ to be a norm of an element in
L if and only if θ is a norm of an element in Lv = Kv ⊗ L for every place v
excluding possibly one.
Proof. The quaternion algebra H = {L, θ} is isomorphic to M(2, K) if and only
if θ ∈ n(L) by I,2.4., and for that if and only if Hv � M(2, K) for every place v
excluding possibly one byu property I and II. Since Hv � {Lv, θ}, the corollary
is proved.