The Arithmetic of Quaternion Algebra

3.3. CASSIFICATION 57
Verify the following numerative table:
ζ(−1) = − 1
2 2 ·3
ζ(−7) = 1
2 4 ·3·5
3.3 Cassification
, ζ(−3) = 1
2 3 ·3·5
, ζ(−5) = − 1
2 2 ·3 2 ·7
, ζ(−9) = 1
3·2 2 ·11 , ζ(−11) = 691
2 3 ·3 2 ·5·7·13
We intend to explain how the classification theorem can be proved by zeta
functions, and how one can deduce from it the reciprocal formula for Hilbert
symbol and the Hasse-Minkowski principle for quadratic forms.
Theorem 3.3.1. (Classification). The number |Ram(H)| of ramified places in
a quaternion algebra H over K is even. For every finite set S of the places of
K with even number S, it exists one and only one quaternion algebra H over
K, up to isomorphism, such that S = Ram(H).
Another equivalent statement of the theorem is formulated by an exact sequence
:
1 −−−−→ Quat(K)
i
−−−−→ ⊕Quat(Kv)
ε
−−−−→ {∓1} −−−−→ 1,
where i is the mapping which assigns an algebra H the set of its localization
modulo isomorphism, and ε is the Hasse invariant: it associates (Hv) with the
product of the Hasse invariants Hv, i.e. −1 if the number of Hv which are fields
is odd, 1 otherwise.
Proof of part of the classification thanks to the zeta functions.
If H is a field, we saw in Theorem 2.2 that ZH(s) has the simple poles at 0 and
1, and is holomorphic elsewhere. The expression ZH in function of ZK which
we recall in (2.1) that
ZH(s/2) = ZK(s)ZK(s − 1)JH(s)
where JH(s) has a zero of order −2 + Ram(H) at point s = 1, shows the order
of ZH at point s = 1/2 equals the order −2 + Ram(H). Then the fundamental
result follows:
Property I.
Characterization of matrix algebra : for H = M(2, K) if and only if Hv = M(2, Kv for every place v.
It follows then (Lam [1], O’Meara [1]):
Corollary 3.3.2. (Hasse-Minkowski principle for quadratic forms). Let q be a
quadratic form over a global field of characteristic unequal 2. Then q is isotropic
over K if and only if q is isotropic over Kv for every place v.
We notice that in the two theorems, one can replaces ”for every place” by ”
for every place possibly excluding someone”
We now explain how the Hasse-Minkowski principle can be derived from the
theorem of the characterization of matrix algebra. Let n be the number of
variables of the quadratic form q.
n = 1, there is nothing to prove.
n = 2, q(x, y) = ax2 +by2 , up equivalence on K, and the principle is equivalent
to the square theorem: a ∈ K× 2 ↔ a ∈ K × 2
v , ∀v. We shall give it a proof by
in advantage of the zeta functions. If L = K( √ a) is isomorphic locally to K ⊕K
.