Klein quartic

In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its automorphism group is isomorphic to PSL(2,7).

Klein's quartic occurs all over mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and the Stark-Heegner theorem on imaginary quadratic number fields of class number one.

As an algebraic curve

The Klein quartic can be viewed as an algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous 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 suitable Fuchsian group Γ(I) which is the principal congruence 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 suitable Hurwitz 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 heptagons. 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 the hyperbolic plane by Möbius transformations, 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.)]