The Arithmetic of Quaternion Algebra

42 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD
vol(Γ(p m ) = D −3/2
K
Np−3m .
where Np is the number of elements of residue field k of K. If O = O0 is
a maximal order of a quaternion algebra H/K, we have
vol(O 1 0) = D −3/2
K (1 − Np=2 �
(Np − 1)
) ·
−1 , if H is a field
1, if H = M(2, K) .
4. (Pitzer [3]).Let {Lnr, p} be the quaternion field over K, uniquely up to
isomorphism. It has the following representation
� �
a b
H = {Lnr, p} = { |a, b ∈ Lnr}
pb a
where 4p is a uniform parameter of K and Lnr/K is a ramified quadratic
extension. We denote simply the above matrix by [a, b]. The order
O2r+1 = {[a, prb]|a, b ∈ RL} is called the canonical order of level Rp2r+1 ,
where RL is the integer ring of Lnr. Verify O2r+1 is actually an order,
and either directly or on the discriminant that O1 is the maximal order.
Verify that an order O is isomorphic to O2r+1 for a r ≥ 0 if and only if it
contains a sub-ring being isomorphic to RL. Prove, if [a, b] ∈ O × 1
, it can
be written as [a, b] = [a ′ , p r b ′ ][1, c], where c = b/amod(p r ) and a ′ , b ′ ∈ RL.
Deduce [O × 1 : O× 2r+1 ] = Np2r .
Deduce the volume of O 1 m for the Tamagawa measure is equal to
vol(O 1 m) = D −3/2
K (1 − Np−2 )(Np − 1) −1 Np 1−m , m ≥ 1.
The formula is a natural generation of that in the above formulae.
5. Maximal compact subgroup. Let K be a non-archimedean local field, and
H/K be a quaternion algebra. Set X = H or K. Prove the maximal
compact subgroups of X × are the unit group B × of the maximal order B
of X.