Английская Википедия:Hurwitz quaternion order
Шаблон:Short description 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 <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 of <math>D</math>, described explicitly by Noam Elkies.[4]
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 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: <math>sys > \frac{4}{3}\log g</math> where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak;[6] see systoles of surfaces.
See also
References
