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).

Contents

• 
  1 Definition
  


• 
  2 Module structure
  


• 
  3 Principal congruence subgroups
  


• 
  4 Application
  


• 
  5 See also
  


• 
  6 References
  

Definition

Let K{displaystyle K} be the maximal real subfield of Q{displaystyle mathbb {Q} }(ρ){displaystyle (rho )} where ρ{displaystyle rho } is a 7th-primitive root of unity.
The ring of integers of K{displaystyle K} is Z[η]{displaystyle mathbb {Z} [eta ]}, where the element η=ρ+ρ¯{displaystyle eta =rho +{bar {rho }}} can be identified with the positive real 2cos⁡(2π7){displaystyle 2cos({tfrac {2pi }{7}})}. Let D{displaystyle D} be the quaternion algebra, or symbol algebra

D:=(η,η)K,{displaystyle D:=,(eta ,eta )_{K},}

so that i2=j2=η{displaystyle i^{2}=j^{2}=eta } and ij=−ji{displaystyle ij=-ji} in D.{displaystyle D.} Also let τ=1+η+η2{displaystyle tau =1+eta +eta ^{2}} and j′=12(1+ηi+τj){displaystyle j'={tfrac {1}{2}}(1+eta i+tau j)}. Let

QHur=Z[η][i,j,j′].{displaystyle {mathcal {Q}}_{mathrm {Hur} }=mathbb {Z} [eta ][i,j,j'].}

Then QHur{displaystyle {mathcal {Q}}_{mathrm {Hur} }} is a maximal order of D{displaystyle D}, described explicitly by Noam Elkies.^[4]

Module structure

The order QHur{displaystyle Q_{mathrm {Hur} }} is also generated by elements

g2=1ηij{displaystyle g_{2}={tfrac {1}{eta }}ij}

and

g3=12(1+(η2−2)j+(3−η2)ij).{displaystyle g_{3}={tfrac {1}{2}}(1+(eta ^{2}-2)j+(3-eta ^{2})ij).}

In fact, the order is a free Z[η]{displaystyle mathbb {Z} [eta ]}-module over
the basis 1,g2,g3,g2g3{displaystyle ,1,g_{2},g_{3},g_{2}g_{3}}. Here the generators satisfy the relations

g22=g33=(g2g3)7=−1,{displaystyle 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⊂Z[η]{displaystyle Isubset mathbb {Z} [eta ]} is by definition the group

QHur1(I)={x∈QHur1:x≡1({displaystyle {mathcal {Q}}_{mathrm {Hur} }^{1}(I)={xin {mathcal {Q}}_{mathrm {Hur} }^{1}:xequiv 1(}mod IQHur)},{displaystyle I{mathcal {Q}}_{mathrm {Hur} })},}

namely, the group of elements of reduced norm 1 in QHur{displaystyle {mathcal {Q}}_{mathrm {Hur} }} equivalent to 1 modulo the ideal IQHur{displaystyle 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>43log⁡g{displaystyle 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.

See also

• (2,3,7) triangle group


• Klein quartic


• Macbeath surface


• First Hurwitz triplet

References

1. 
   ^ Vogeler, Roger (2003), On the geometry of Hurwitz surfaces, PhD thesis, Florida State University.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.
   


2. 
   ^ 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.
   


3. 
   ^ 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.
   


4. 
   ^ 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.
   


5. 
   ^ 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.
   


6. 
   ^ Buser, P.; Sarnak, P. (1994), "On the period matrix of a Riemann surface of large genus", Inventiones Mathematicae, 117 (1): 27–56, Bibcode:1994InMat.117...27B, doi:10.1007/BF01232233, MR 1269424. With an appendix by J. H. Conway and N. J. A. Sloane.
   

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