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 $\,1,g_{2},g_{3},g_{2}g_{3}$ . Here the generators satisfy the relations

g_{2}^{2}=g_{3}^{3}=(g_{2}g_{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}}_{\mathrm {Hur} }^{1}(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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