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

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.^[4]

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^[5] 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;^[6] see systoles of surfaces.

References

↑ Vogeler, Roger (2003), On the geometry of Hurwitz surfaces, PhD thesis, Florida State University .
↑ 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 .
↑ 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 .
↑ 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 .
↑ 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 .
↑ Buser, P.; Sarnak, P. (1994), "On the period matrix of a Riemann surface of large genus", Inventiones Mathematicae 117 (1): 27–56, doi:10.1007/BF01232233, MR 1269424. With an appendix by J. H. Conway and N. J. A. Sloane.

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