# 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

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.

## See also

## References

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