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).


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


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).


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


  1. Vogeler, Roger (2003), On the geometry of Hurwitz surfaces, PhD thesis, Florida State University.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
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