In the mathematical 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 ver ...

s, the first Hurwitz triplet is a triple of distinct

Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by virt ...

s with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, respectively the

Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...

and the Macbeath surface). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, 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 would be invertible 2 × 2 integer matrices of determinant 1, in which the ...

s defined by the triplet of primes produce

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...

s corresponding to the triplet of Riemann surfaces.

Arithmetic construction

Let

K

be the real subfield of

\mathbb

rho Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...

/math> where

\rho

is a 7th-primitive

root of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...

. 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 deno ...

of ''K'' is

\mathbb

eta Eta (uppercase , lowercase ; grc, ἦτα ''ē̂ta'' or ell, ήτα ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel . Originally denoting the voiceless glottal fricative in most dialects, ...

/math>, where

\eta=2\cos(\tfrac)

. Let

D

be the

quaternion algebra In mathematics, a quaternion algebra over a 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 4 over ''F''. Every quaternion algebra becomes a ma ...

, or symbol algebra

(\eta,\eta)_

. Also Let

\tau=1+\eta+\eta^2

and

j'=\tfrac(1+\eta i + \tau j)

. Let

\mathcal_\mathrm=\mathbb

i,j,j']. Then

\mathcal_\mathrm

is a maximal Order (ring theory), order of

D

(see

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. ...

), described explicitly by

Noam Elkies Noam David Elkies (born August 25, 1966) is a professor of mathematics at Harvard University. At the age of 26, he became the youngest professor to receive tenure at Harvard. He is also a pianist, chess national master and a chess composer. E ...

In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in

\mathbb

/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 In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...

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) In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: : \mbox(2,\mathbf) = \left\. It is a connected non-compact simple real Lie group of dimension 3 wit ...

. 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 ' ...

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 to its underlying topology. In the simplest application, the case of a t ...

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 spac ...

of the surface and

K(u)

is the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...

. 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. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ...

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.

References

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

Arithmetic construction

Bound for systolic length and the systolic ratio

See also

References