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In the
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
theory of
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 ...
s, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, respectively the Klein quartic and the Macbeath surface). The explanation for this phenomenon is arithmetic. Namely, in 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 the appropriate
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 rational prime 13 splits as a product of three distinct prime ideals. The principal
congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible integer matrices of determinant 1 in which the off-diag ...
s defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces.


Arithmetic construction

Let K be the real subfield of \mathbb
rho Rho (; uppercase Ρ, lowercase ρ or ; or ) is the seventeenth letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician alphabet, Phoenician letter resh . Its uppercase form uses the same ...
/math> 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 \eta=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 (\eta,\eta)_. Also Let \tau=1+\eta+\eta^2 and j'=\tfrac(1+\eta i + \tau j). Let \mathcal_\mathrm=\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_\mathrm is a maximal Order (ring theory), order of D (see Hurwitz quaternion order), described explicitly by Noam Elkies In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in \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>, namely :13=\eta (\eta +2)(2\eta-1)(3-2\eta)(\eta+3), where \eta (\eta+2) is invertible. Also consider the prime ideals generated by the non-invertible factors. The principal congruence subgroup defined by such a prime ideal ''I'' is by definition the group :\mathcal^1_\mathrm(I) = \, namely, the group of elements of reduced norm 1 in \mathcal_\mathrm 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). Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a Fuchsian model, the quotient of the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
by one of these three Fuchsian groups.


Bound for systolic length and the systolic ratio

The
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a Surface (topology), surface to its underlying topology. In the simplest applicati ...
states that :\chi(\Sigma)=\frac \int_ K(u)\,dA, where \chi(\Sigma) is the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of the surface and K(u) is the
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
. In the case g=14 we have :\chi(\Sigma)=-26 and K(u)=-1, thus we obtain that the area of these surfaces is :52\pi. The lower bound on the
systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...
as specified in namely :\frac \log(g(\Sigma)), is 3.5187. Some specific details about each of the surfaces are presented in the following tables (the number of systolic loops is taken from . The term Systolic Trace refers to the least reduced trace of an element in the corresponding subgroup \mathcal^1_(I). The systolic ratio is the ratio of the square of the systole to the area.


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 ...


References

* * *{{cite book , last=Vogeler , first=R. , title=On the geometry of Hurwitz surfaces , type=Thesis , publisher=Florida State University , year=2003 Riemann surfaces Differential geometry of surfaces Systolic geometry