- Klein quartic
In
hyperbolic geometry , the Klein quartic, named afterFelix Klein , is a compactRiemann surface of genus 3 with the highest possible orderautomorphism group for this genus, namely order 168. As such, the Klein quartic is theHurwitz surface of lowest possible genus; seeHurwitz's automorphisms theorem . Its automorphism group is isomorphic toPSL(2,7) .Klein's quartic occurs all over mathematics, in contexts including
representation theory ,homology theory ,octonion multiplication ,Fermat's last theorem , and theStark-Heegner theorem onimaginary quadratic number field s of class number one.As an algebraic curve
The Klein quartic can be viewed as an
algebraic curve over thecomplex number s C, defined by the following quartic equation inhomogeneous coordinates ::"x"3"y" + "y"3"z" + "z"3"x" = 0.Quaternion algebra construction
The compact Klein quartic can be constructed as the quotient of the
hyperbolic plane by the action of a suitableFuchsian group Γ(I) which is the principalcongruence subgroup associated with the ideal I=langle eta-2 angle in the ring of integers mathbb{Z}(eta) of the field mathbb Q(eta) where eta=2cos (2pi/7). Note the identity:2-eta)^3= 7(eta-1)^2,
exhibiting 2-eta as a prime factor of 7 in the ring of integers.
The group Γ(I) is a subgroup of the (2,3,7) hyperbolic
triangle group . Namely, Γ(I) is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the generators i,j and relations i^2=j^2=eta, quad ij=-ji. One chooses a suitableHurwitz quaternion order mathcal Q_{mathrm{Hur in the quaternion algebra, Γ(I) is then the group of norm 1 elements in 1+Imathcal Q_{mathrm{Hur. The least absolute value of a trace of a hyperbolic element in Γ(I) is eta^2+3eta+2, corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this genus.Tiling
The Klein quartic can be tiled with 24 regular hyperbolic
heptagon s. The order of the automorphism group is thus related to the fact that:24 × 7 = 168.Considering the action of SL(2,R) on the upper half-plane model H2 of thehyperbolic plane byMöbius transformation s, the affine Klein quartic can be realized as the quotient Γ(7)H2. (Here Γ(7) is the subgroup of SL(2,Z) consisting of matrices that are congruent to the identity matrix when all entries are taken modulo 7.)ee also
*
Bolza surface
*Macbeath surface
*First Hurwitz triplet References
*Elkies, N.: Shimura curve computations. Algorithmic number theory (Portland, OR, 1998), 1–47, Lecture Notes in Computer Science, 1423, Springer, Berlin, 1998. See arXiv|math.NT|0005160
External links
* [http://www.math.uni-siegen.de/wills/klein/ Polyhedral models of Felix Klein's quartic]
* [http://www.msri.org/publications/books/Book35/contents.html "The Eightfold Way: The Beauty of Klein's Quartic Curve" (Silvio Levy, ed.)]
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