Hurwitz quaternion order

The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces.^[1] The Hurwitz quaternion order was studied in 1967 by Goro Shimura,^[2] but first explicitly described by Noam Elkies in 1998.^[3] For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature).

Definition

Let $K$ be the maximal real subfield of $\mathbb{Q}$ $(\rho)$ where $\rho$ is a 7th-primitive root of unity. The ring of integers of $K$ is $\mathbb{Z}[\eta]$ , where the element $\eta=\rho+ \bar\rho$ can be identified with the positive real $2\cos(\tfrac{2\pi}{7})$ . Let $D$ be the quaternion algebra, or symbol algebra

D:=\,(\eta,\eta)_{K},

so that $i^2=j^2=\eta$ and $ij=-ji$ in $D.$ Also let $\tau=1+\eta+\eta^2$ and $j'=\tfrac{1}{2}(1+\eta i + \tau j)$ . Let

\mathcal{Q}_{\mathrm{Hur}}=\mathbb{Z}[\eta][i,j,j'].

Then $\mathcal{Q}_{\mathrm{Hur}}$ is a maximal order of $D$ , described explicitly by Noam Elkies.^[4]

Module structure

The order $Q_{\mathrm{Hur}}$ is also generated by elements

g_2= \tfrac{1}{\eta}ij

and

g_3=\tfrac{1}{2}(1+(\eta^2-2)j+(3-\eta^2)ij).

In fact, the order is a free $\mathbb Z[\eta]$ -module over the basis $\,1,g_2,g_3, g_2g_3$ . Here the generators satisfy the relations

g_2^2=g_3^3= (g_2g_3)^7=-1,

which descend to the appropriate relations in the (2,3,7) triangle group, after quotienting by the center.

Principal congruence subgroups

The principal congruence subgroup defined by an ideal $I \subset \mathbb{Z}[\eta]$ is by definition the group

\mathcal{Q}^1_{\mathrm{Hur}}(I) = \{x \in \mathcal{Q}_{\mathrm{Hur}}^1 : x \equiv 1 (

mod

I\mathcal{Q}_{\mathrm{Hur}})\},

namely, the group of elements of reduced norm 1 in $\mathcal{Q}_{\mathrm{Hur}}$ equivalent to 1 modulo the ideal $I\mathcal{Q}_{\mathrm{Hur}}$ . The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to PSL(2,R).

Application

The order was used by Katz, Schaps, and Vishne^[5] to construct a family of Hurwitz surfaces satisfying an asymptotic lower bound for the systole: $sys > \frac{4}{3}\log g$ where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak;^[6] see systoles of surfaces.

References