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

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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

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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

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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).

Application

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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

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References

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  1. ^ Vogeler, Roger (2003), On the geometry of Hurwitz surfaces (PhD), Florida State University.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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.
  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, doi:10.4310/jdg/1180135693, MR 2331526, S2CID 18152345.
  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, S2CID 116904696. With an appendix by J. H. Conway and N. J. A. Sloane.{{citation}}: CS1 maint: postscript (link)