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

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

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

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

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

Module structure

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

${\displaystyle g_{2}={\tfrac {1}{\eta ))ij}$

and

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

In fact, the order is a free ${\displaystyle \mathbb {Z} [\eta ]}$-module over the basis ${\displaystyle \,1,g_{2},g_{3},g_{2}g_{3))$. Here the generators satisfy the relations

${\displaystyle g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=-1,}$

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 ${\displaystyle I\subset \mathbb {Z} [\eta ]}$ is by definition the group

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

namely, the group of elements of reduced norm 1 in ${\displaystyle {\mathcal {Q))_{\mathrm {Hur} ))$ equivalent to 1 modulo the ideal ${\displaystyle I{\mathcal {Q))_{\mathrm {Hur} ))$. 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: ${\displaystyle sys>{\frac {4}{3))\log g}$ where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak;^[6] see systoles of surfaces.