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
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 subgroupsedit
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).
Applicationedit
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.
^Vogeler, Roger (2003), On the geometry of Hurwitz surfaces (PhD), Florida State University.
^Shimura, Goro (1967), "Construction of class fields and zeta functions of algebraic curves", Annals of Mathematics, Second Series, 85 (1): 58–159, doi:10.2307/1970526, JSTOR 1970526, MR 0204426.
^Elkies, Noam D. (1998), "Shimura curve computations", Algorithmic number theory (Portland, OR, 1998), Lecture Notes in Computer Science, vol. 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" (PDF), in Levi, Sylvio (ed.), The Eightfold Way: The Beauty of Klein's Quartic Curve, Mathematical Sciences Research Institute publications, vol. 35, Cambridge University 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, doi:10.4310/jdg/1180135693, MR 2331526, S2CID 18152345.
^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, S2CID 116904696. With an appendix by J. H. Conway and N. J. A. Sloane.{{citation}}: CS1 maint: postscript (link)