Hurwitz quaternion order

Definition

Let K be the maximal real subfield of Q ( ρ ) where ρ is a 7th-primitive root of unity. The ring of integers of K is Z [ η ] , where the element η = ρ + ρ ¯ can be identified with the positive real 2 cos ⁡ ( 2 π 7 ) . Let D be the quaternion algebra, or symbol algebra

so that i 2 = j 2 = η and i j = − j i in D . Also let τ = 1 + η + η 2 and j ′ = 1 2 ( 1 + η i + τ j ) . Let

Then Q H u r is a maximal order of D , described explicitly by Noam Elkies.

Module structure

The order Q H u r is also generated by elements

and

In fact, the order is a free Z [ η ] -module over the basis 1 , g 2 , g 3 , g 2 g 3 . 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 I ⊂ Z [ η ] is by definition the group

namely, the group of elements of reduced norm 1 in Q H u r equivalent to 1 modulo the ideal I Q H u r . 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: s y s > 4 3 log ⁡ g where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak; see systoles of surfaces.