Hurwitz quaternion order
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The Hurwitz quaternion order is a specific
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
in a
quaternion algebra In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
over a suitable
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
. The order is of particular importance in
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
theory, in connection with surfaces with maximal
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967 by
Goro Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multip ...
, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see
Hurwitz quaternion In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are ''either'' all integers ''or'' all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz ...
(both usages are current in the literature).


Definition

Let K be the maximal real subfield of \mathbb(\rho) where \rho is a 7th-primitive
root of unity In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
. The
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
of K is \mathbb
eta Eta ( ; uppercase , lowercase ; ''ē̂ta'' or ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel, . Originally denoting the voiceless glottal fricative, , in most dialects of Ancient Greek, it ...
/math>, where the element \eta=\rho+ \bar\rho can be identified with the positive real 2\cos(\tfrac). Let D be the
quaternion algebra In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
, or symbol algebra :D:=\,(\eta,\eta)_, so that i^2=j^2=\eta and ij=-ji in D. Also let \tau=1+\eta+\eta^2 and j'=\tfrac(1+\eta i + \tau j). Let :\mathcal_=\mathbb
eta Eta ( ; uppercase , lowercase ; ''ē̂ta'' or ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel, . Originally denoting the voiceless glottal fricative, , in most dialects of Ancient Greek, it ...
i,j,j']. Then \mathcal_ is a maximal Order (ring theory), order of D, described explicitly by Noam Elkies.


Module structure

The order Q_ is also generated by elements :g_2= \tfracij and :g_3=\tfrac(1+(\eta^2-2)j+(3-\eta^2)ij). In fact, the order is a free \mathbb Z
eta Eta ( ; uppercase , lowercase ; ''ē̂ta'' or ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel, . Originally denoting the voiceless glottal fricative, , in most dialects of Ancient Greek, it ...
/math>-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 In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, 84(''g'' − 1), of it ...
, after quotienting by the center.


Principal congruence subgroups

The principal congruence subgroup defined by an ideal I \subset \mathbb
eta Eta ( ; uppercase , lowercase ; ''ē̂ta'' or ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel, . Originally denoting the voiceless glottal fricative, , in most dialects of Ancient Greek, it ...
/math> is by definition the group :\mathcal^1_(I) = \, namely, the group of elements of reduced norm 1 in \mathcal_ equivalent to 1 modulo the ideal I\mathcal_. The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to P SL(2,R).


Application

The order was used by Katz, Schaps, and Vishne to construct a family of Hurwitz surfaces satisfying an asymptotic lower bound for the systole: sys > \frac\log g where g is the genus, improving an earlier result of Peter Buser and Peter Sarnak; see systoles of surfaces.


See also

*
(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, 84(''g'' − 1), of it ...
* Klein quartic * Macbeath surface * First Hurwitz triplet


References

{{reflist, 30em Riemann surfaces Differential geometry of surfaces Algebras
Algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
Systolic geometry