# 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 be the maximal real subfield of where is a 7th-primitive root of unity. The ring of integers of is , where the element can be identified with the positive real . Let be the quaternion algebra, or symbol algebra

so that and in Also let and . Let

Then is a maximal order of , described explicitly by Noam Elkies.[4]

## Module structure

The order is also generated by elements

and

In fact, the order is a free -module over the basis . Here the generators satisfy the relations

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 is by definition the group

- mod

namely, the group of elements of reduced norm 1 in equivalent to 1 modulo the ideal . 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: where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak;[6] see systoles of surfaces.

## References

- Vogeler, Roger (2003),
*On the geometry of Hurwitz surfaces*, PhD thesis, Florida State University. - Shimura, Goro (1967), "Construction of class fields and zeta functions of algebraic curves",
*Annals of Mathematics*, Second Series,**85**: 58–159, doi:10.2307/1970526, MR 0204426. - Elkies, Noam D. (1998), "Shimura curve computations",
*Algorithmic number theory (Portland, OR, 1998)*, Lecture Notes in Computer Science,**1423**, Berlin: Springer-Verlag, pp. 1–47, arXiv:math.NT/0005160, doi:10.1007/BFb0054850, MR 1726059. - Elkies, Noam D. (1999), "The Klein quartic in number theory",
*The eightfold way*, Math. Sci. Res. Inst. Publ.,**35**, Cambridge: Cambridge Univ. Press, pp. 51–101, MR 1722413. - Katz, Mikhail G.; Schaps, Mary; Vishne, Uzi (2007), "Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups",
*Journal of Differential Geometry*,**76**(3): 399–422, arXiv:math.DG/0505007, MR 2331526. - Buser, P.; Sarnak, P. (1994), "On the period matrix of a Riemann surface of large genus",
*Inventiones Mathematicae*,**117**(1): 27–56, Bibcode:1994InMat.117...27B, doi:10.1007/BF01232233, MR 1269424. With an appendix by J. H. Conway and N. J. A. Sloane.