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. The Hurwitz quaternion order was studied in '67 by Goro Shimura [4] , but first explicitly described by Noam Elkies in '98. For an alternative use of the term, see Integer quaternion (both usages are current in the literature).

Definition

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

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

so that $i^2=j^2=eta,;\; ij=-ji$ in $D.$ Also let $au=1+eta+eta^2$ and $j\text{'}=\; frac\{1\}\{2\}(1+eta\; i\; +\; au\; j)$. Let

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

Then $mathcal\{Q\}\_\{mathrm\{Hur$ is a maximal order of $D$, described explicitly by Noam Elkies [1] .

Module structure

The order $Q\_\{mathrm\{Hur$ is also generated by elements

:$g\_2=\; frac\{1\}\{eta\}ij$

and

:$g\_3=\; frac\{1\}\{2\}(1+(eta^2-2)j+(3-eta^2)ij).$

In fact, the order is a free $mathbb\; Z\; [eta]$-module overthe 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 $Imathcal\{Q\}\_\{mathrm\{Hur)\}$,

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

ee also

*(2,3,7) triangle group
*Klein quartic
*Macbeath surface
*First Hurwitz triplet

References

* [1] Elkies, N.: The Klein quartic in number theory. The eightfold way, 51– 101, Math. Sci. Res. Inst. Publ. 35, Cambridge Univ. Press, Cambridge, 1999.

* [2] Elkies, N.: Shimura curve computations. "Algorithmic number theory" (Portland, OR, 1998), 1–47, Lecture Notes in Computer Science, 1423, Springer, Berlin, 1998. See arXiv|math.NT|0005160

* [3] Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), 399-422. Available at arXiv:math.DG/0505007.

* [4] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. of Math. (2) 85 (1967), 58--159.

* [5] Vogeler, R.: On the geometry of Hurwitz surfaces. Thesis. Florida State University. 2003.