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.{{citation

| last = Vogeler | first = Roger

| publisher = Florida State University

| type = PhD | url=http://purl.flvc.org/fsu/fd/FSU_migr_etd-4544

| title = On the geometry of Hurwitz surfaces

| year = 2003}}. The Hurwitz quaternion order was studied in 1967 by Goro Shimura,{{citation

| last = Shimura | first = Goro | author-link = 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| issue = 1 | jstor = 1970526 }}. but first explicitly described by Noam Elkies in 1998.{{citation

| last = Elkies | first = Noam D. | author-link = Noam Elkies

| contribution = Shimura curve computations

| doi = 10.1007/BFb0054850 | doi-access = free

| 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}}. For an alternative use of the term, see Hurwitz quaternion (both usages are current in the literature).

Definition

Let $K$ be the maximal real subfield of $\mathbb{Q}$ $(\rho)$ where $\rho$ is a 7th-primitive root of unity.

The ring of integers of $K$ is $\mathbb{Z}[\eta]$ , where the element $\eta=\rho+ \bar\rho$ can be identified with the positive real $2\cos(\tfrac{2\pi}{7})$ . Let $D$ be the quaternion algebra, or symbol algebra

: $D:=\,(\eta,\eta)_{K},$

so that $i^2=j^2=\eta$ and $ij=-ji$ in $D.$ Also let $\tau=1+\eta+\eta^2$ and $j'=\tfrac{1}{2}(1+\eta i + \tau j)$ . Let

: $\mathcal{Q}_{\mathrm{Hur}}=\mathbb{Z}[\eta][i,j,j'].$

Then $\mathcal{Q}_{\mathrm{Hur}}$ is a maximal order of $D$ , described explicitly by Noam Elkies.{{citation

| last = Elkies | first = Noam D. | author-link = Noam Elkies

| contribution = The Klein quartic in number theory

| mr = 1722413

| pages = 51–101

| chapter-url=http://library.msri.org/books/Book35/files/elkies.pdf

| editor-first=Sylvio |editor-last=Levi

| title=The Eightfold Way: The Beauty of Klein's Quartic Curve

| publisher=Cambridge University Press

| series=Mathematical Sciences Research Institute publications

| url=http://library.msri.org/books/Book35/contents.html

| volume = 35

| year = 1999}}.

Module structure

The order $Q_{\mathrm{Hur}}$ is also generated by elements

: $g_2= \tfrac{1}{\eta}ij$

and

: $g_3=\tfrac{1}{2}(1+(\eta^2-2)j+(3-\eta^2)ij).$

In fact, the order is a free $\mathbb Z[\eta]$ -module over

the basis $\,1,g_2,g_3, g_2g_3$ . Here the generators satisfy the relations

: $g_2^2=g_3^3= (g_2g_3)^7=-1,$

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 $I \subset \mathbb{Z}[\eta]$ is by definition the group

: $\mathcal{Q}^1_{\mathrm{Hur}}(I) = \{x \in \mathcal{Q}_{\mathrm{Hur}}^1 : x \equiv 1 ($ mod $I\mathcal{Q}_{\mathrm{Hur}})\},$

namely, the group of elements of reduced norm 1 in $\mathcal{Q}_{\mathrm{Hur}}$ equivalent to 1 modulo the ideal $I\mathcal{Q}_{\mathrm{Hur}}$ . 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{{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| doi = 10.4310/jdg/1180135693 | s2cid = 18152345 }}. to construct a family of Hurwitz surfaces satisfying an asymptotic lower bound for the systole: $sys > \frac{4}{3}\log g$ where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak;{{citation

| last1 = Buser | first1 = P. | authorlink1=Jürg Peter Buser

| last2 = Sarnak | first2 = P. | authorlink2=Peter Sarnak

| doi = 10.1007/BF01232233 | doi-access=

| 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.| bibcode = 1994InMat.117...27B

| s2cid = 116904696 }} see systoles of surfaces.