Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q

Question

I'm looking for an explanation of the following result:

If D is a maximal order in a definite (i.e. ramified at infinity) quaternion algebra B over $\mathbb{Q}$, and $\phi : B\otimes\mathbb{R}\rightarrow\mathbb{H}$ is an isomorphism over $\mathbb{R}$ (where $\mathbb{H}$ is the Hamiltonian quaternions over $\mathbb{R}$), then the co-volume of $\phi(D)$ in $\mathbb{H}$ (with respect to the Lebesgue measure for the basis 1,i,j,k) is $(1/4) \cdot Disc(D)$ (where the discriminant is the determinant of the trace form on a basis of $D$, or, equivalently since $D$ is maximal, the product of the finite primes where $B$ ramifies).

Right now the only way I have to see this is through an explicit calculation using the generators for maximal orders described in a paper by Ibukiyama available here: Link , but I feel like there must be a nice way to see this using just properties of maximal orders without having to calculate with an explicit basis. Any ideas or references?

Answer 1

As in the Iwasawa-Tate style treatment of zeta functions of number fields and residue of first pole as volume of idele class group (with the non-compact "ray" removed), this volume is essentially the residue of the leading pole of the zeta function of the quaternion algebra, and this zeta function factors as zeta and a shift of zeta of the groundfield, up to finitely-many factors depending on ramification of the quaternion algebra.

This kind of computation is treated in Weil's "Basic Number Theory", and also in his "Adeles and Algebraic Groups".

Answer 2

See chapters II and III of the classic book of Vigneras. Beware however that in the later parts of chapter III she assumes widely that C.E. (the "Eichler condition") is verified, and C.E. is explicitly not verified for quaternion algebras $B$ over $\mathbf{Q}$ such that $B\otimes \mathbf{R} \cong \mathbb{H}$ (so called "definite" algebras). For lots of computation with definite algebras please see Chapter V of her book.