The Arithmetic of Quaternion Algebra

4.3. EXAMPLES AND APPLICATIONS 99
follows that Q(O 1 ) is a central simple algebra over k of dimension 4. It is simple
because that by performing tensor product of it with K over k, it becomes
simple. Therefore Q(O) is a quaternion algebra over k. an infinite place w of
k which is extended to a place v of K ramified in H is definitely ramified in
Q(O). An infinite place w of k which is unramified in Q(O 1 ) has their every
extension v in K to be unramified in H. A place w associating with a real
inclusion iw : k → R can be extended in [K : k] real places. It deduces from 1.1
that k = K.
Every isometry of ¯ Γ\H2 to ¯ Γ ′ \H2 is lifted to an isometry of the universal covering
H2. The isometries of H2 forms a group which is isomorphic to P GL(2, R).
It follows that ¯ Γ\H2 and ¯ Γ ′ \H2 are isomorphic if and only if Γ and Γ ′ are
conjugate in GL(2, R). From it Q(σ(O 1 )) and Q(σ ′ (O ,1 )) are conjugate in
GL(2, R). The center remains fixed, thus σ(K) = σ ′ (K ′ ). The quaternion algebra
Q(σ(O 1 )) and Q(σ ′ (O ′ 1 )) are hence isomorphic. It may then suppose
they are equal. Every automorphism of a quaternion algebra is inner one,
therefore there exists a ∈ H × such that σ ′ (O ′ ) = σ(aOa −1 ). We finally have
H ′ = σ ′ −1 σ(H) and O ′ = σ ′ −1σ(aOa −1 ).
It is clear that this proof can be generalized to riemanian manifold ΓX where
X is a product of H2 and H3, and ¯ Γ here is the image of a quaternion group.
The isometric group of X is determined in virtue of a theorem of de Rham [1].
Corollary 4.3.6. If the type number of the order of H is great than the degree
[K : Q], then it exists in H two maximal orders O and O ′ such that the surfaces
¯Γ\H and ¯ Γ ′ \H are isospectral but not isometric.
In fact, the number of the conjugate σ(H) of H is dominated by the degree
[K : Q]. The corollary can be refined considerably, if necessary, by observing:
– the non-maximal orders
– a better upper bound of Card{σ(H)}, depending on the given (K, H).
EXAMPLE. Suppose that K is a real quadratic field, of which the class number
is great than or equal to 4, for example Q( √ 82). Suppose that Ram(H) is
consists of just one infinite place and of the finite places such that all of their
correspondent prime ideals are principal. Therefore, it exists at least 4 types
of maximal orders, and we can construct some isospectral riemannian surfaces
but they are not isometric. We can thus compute the genera of the obtained
surfaces, by the genus formula and the tables of ζK(−1) calculated by Cohen
[1].
EXAMPLE. H is the quaternion field over K = Q( √ 10) which is ramified at
an infinite place, and above the principal prime ideals (7), (11), (11 + 3 √ 10) H
does not contain the roots of unit other than ∓1 since (7) is decomposed into
two cyclotomic quadratic extensions of K, i.e. K( √ −1) and K( √ −3). H is
never fixed by any Q-automorphism and contains two types of maximal orders,
because the class number of Q( √ 10) is 2. The unit groups of reduced norm 1
of two non-equivalent maximal orders allow us to construct two isospectral but
non-isometric surfaces.
Remark 4.3.7. The construction can be generalized and is allowed to construct
some isospectral but non isometric riemannian surfaces in all the dimension
n ≥ 2.