Concept in mathematics
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 1967 by Goro Shimura, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature).
Definition.
Let formula_1 be the maximal real subfield of formula_2formula_3 where formula_4 is a 7th-primitive root of unity. 
The ring of integers of formula_1 is formula_6, where the element formula_7 can be identified with the positive real formula_8. Let formula_9 be the quaternion algebra, or symbol algebra 
formula_10
so that formula_11 and formula_12 in formula_13 Also let formula_14 and formula_15. Let 
formula_16
Then formula_17 is a maximal order of formula_9, described explicitly by Noam Elkies.
Module structure.
The order formula_19 is also generated by elements
formula_20
and
formula_21
In fact, the order is a free formula_22-module over
the basis formula_23. Here the generators satisfy the relations
formula_24
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 formula_25 is by definition the group
formula_26mod formula_27
namely, the group of elements of reduced norm 1 in formula_17 equivalent to 1 modulo the ideal formula_29. 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 to construct a family of Hurwitz surfaces satisfying an asymptotic lower bound for the systole: formula_30 where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak; see systoles of surfaces.
References.
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