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The '''Hurwitz quaternion order''' is a specific [[order (ring theory)|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 surface]]s.<ref>{{citation
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| last = Vogeler | first = Roger
| publisher = Florida State University
| series = PhD thesis
| title = On the geometry of Hurwitz surfaces
| year = 2003}}.</ref>  The Hurwitz quaternion order was studied in 1967 by [[Goro Shimura]],<ref>{{citation
| last = Shimura | first = Goro | authorlink = Goro Shimura
| doi = 10.2307/1970526
| mr = 0204426
| journal = [[Annals of Mathematics]] | series = Second Series
| pages = 58–159
| title = Construction of class fields and zeta functions of algebraic curves
| volume = 85
| year = 1967}}.</ref> but first explicitly described by [[Noam Elkies]] in 1998.<ref>{{citation
| last = Elkies | first = Noam D. | authorlink = Noam Elkies
| contribution = Shimura curve computations
| doi = 10.1007/BFb0054850
| arxiv = math.NT/0005160
| mr = 1726059
| location = Berlin
| pages = 1–47
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Algorithmic number theory (Portland, OR, 1998)
| volume = 1423
| year = 1998}}.</ref> For an alternative use of the term, see [[Hurwitz quaternion]] (both usages are current in the literature).


==Definition==
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Let <math>K</math> be the maximal real subfield of <math>\mathbb{Q}</math><math>(\rho)</math> where <math>\rho</math> is a 7th-primitive [[root of unity]].  
The [[ring of integers]] of <math>K</math> is <math>\mathbb{Z}[\eta]</math>, where the element <math>\eta=\rho+ \bar\rho</math> can be identified with the positive real <math>2\cos(\tfrac{2\pi}{7})</math>. Let <math>D</math> be the [[quaternion algebra]], or symbol algebra
 
:<math>D:=\,(\eta,\eta)_{K},</math>
 
so that <math>i^2=j^2=\eta</math> and <math>ij=-ji</math> in <math>D.</math> Also let <math>\tau=1+\eta+\eta^2</math> and <math>j'=\tfrac{1}{2}(1+\eta i + \tau j)</math>. Let
 
:<math>\mathcal{Q}_{\mathrm{Hur}}=\mathbb{Z}[\eta][i,j,j'].</math>
 
Then <math>\mathcal{Q}_{\mathrm{Hur}}</math> is a maximal [[Order (ring theory)|order]] of <math>D</math>, described explicitly by [[Noam Elkies]].<ref>{{citation
  | last = Elkies | first = Noam D. | authorlink = Noam Elkies
| contribution = The Klein quartic in number theory
| mr = 1722413
| location = Cambridge
| pages = 51–101
| publisher = Cambridge Univ. Press
| series = Math. Sci. Res. Inst. Publ.
| title = The eightfold way
| volume = 35
| year = 1999}}.</ref>
 
==Module structure==
The order <math>Q_{\mathrm{Hur}}</math> is also generated by elements
 
:<math>g_2= \tfrac{1}{\eta}ij</math>
 
and
 
:<math>g_3=\tfrac{1}{2}(1+(\eta^2-2)j+(3-\eta^2)ij).</math>
 
In fact, the order is a free <math>\mathbb Z[\eta]</math>-module over
the basis <math>\,1,g_2,g_3, g_2g_3</math>. Here the generators satisfy the relations
 
:<math>g_2^2=g_3^3= (g_2g_3)^7=-1,</math>
 
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 <math>I \subset \mathbb{Z}[\eta]</math> is by definition the group
 
:<math>\mathcal{Q}^1_{\mathrm{Hur}}(I) = \{x \in \mathcal{Q}_{\mathrm{Hur}}^1 : x \equiv 1  (</math>mod <math>I\mathcal{Q}_{\mathrm{Hur}})\},</math>
 
namely, the group of elements of [[reduced norm]] 1 in <math>\mathcal{Q}_{\mathrm{Hur}}</math> equivalent to 1 modulo the ideal <math>I\mathcal{Q}_{\mathrm{Hur}}</math>. The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to P[[SL(2,R)]].
==Application==
The order was used by Katz, Schaps, and Vishne<ref>{{citation
| last1 = Katz | first1 = Mikhail G. | author1-link = Mikhail Katz
| last2 = Schaps | first2 = Mary
| last3 = Vishne | first3 = Uzi
  | author3-link = Uzi Vishne
| arxiv = math.DG/0505007
| mr = 2331526
| issue = 3
| journal = Journal of Differential Geometry
| pages = 399–422
  | title = Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups
| url = http://projecteuclid.org/getRecord?id=euclid.jdg/1180135693
| volume = 76
| year = 2007}}.</ref> to construct a family of Hurwitz surfaces satisfying an asymptotic lower bound for the systole: <math>sys > \frac{4}{3}\log g</math> where g is the genus, improving an earlier result of [[Peter Buser]] and [[Peter Sarnak]];<ref>{{citation
| last1 = Buser | first1 = P.
| last2 = Sarnak | first2 = P.
| doi = 10.1007/BF01232233
| mr = 1269424
| issue = 1
| journal = Inventiones Mathematicae
| pages = 27–56
| title = On the period matrix of a Riemann surface of large genus
| volume = 117
| year = 1994
| postscript = . With an appendix by J. H. Conway and N. J. A. Sloane.}}</ref> see [[systoles of surfaces]].
 
==See also==
*[[(2,3,7) triangle group]]
*[[Klein quartic]]
*[[Macbeath surface]]
*[[First Hurwitz triplet]]
 
==References==
{{reflist|30em}}
 
[[Category:Riemann surfaces]]
[[Category:Differential geometry of surfaces]]
[[Category:Algebras]]
[[Category:Quaternions|Algebra]]
[[Category:Systolic geometry]]

Latest revision as of 01:30, 25 September 2014

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