Macbeath surface

Macbeath surface

In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface.

The automorphism group of the Macbeath surface is the simple group PSL(2,8), consisting of 504 symmetries.harvtxt|Wohlfahrt|1985.]

Triangle group construction

The surface's Fuchsian group can be constructed as the principal congruence subgroup of the (2,3,7) triangle group in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing the ideal langle 2 angle in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its systole is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations.

Historical note

This surface was originally discovered by harvtxt|Fricke|1899, but named after Alexander M. Macbeath due to his later independent rediscovery of the same curve. [harvtxt|Macbeath|1965.] Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed by Serre in a 24.vii.1990 letter to Abhyankar". [harvtxt|Elkies|1998.]

Notes

References

*citation|last1=Berry|first1=Kevin|last2=Tretkoff|first2=Marvin|contribution=The period matrix of Macbeath's curve of genus seven|title=Curves, Jacobians, and abelian varieties, Amherst, MA, 1990|pages=31–40|publisher=Contemp. Math., 136, Amer. Math. Soc.|location=Providence, RI|year=1992|id=MathSciNet|id=1188192.

*citation|last1=Bujalance|first1=Emilio|last2=Costa|first2=Antonio F.|contribution=Study of the symmetries of the Macbeath surface|format=In Spanish|title=Mathematical contributions|pages=375–385|publisher=Editorial Complutense|location=Madrid|year=1994|id=MathSciNet|id=1303808.

*citation|last=Elkies|first=N. D.|authorlink=Noam Elkies|contribution=Shimura curve computations|title=Algorithmic Number Theory: Third International Symposium, ANTS-III|publisher=Springer-Verlag, Lecture Notes in Computer Science 1423|year=1998|doi=10.1007/BFb0054849|id=arxiv|math.NT|0005160|pages=1–47.

*citation|last=Fricke|first=R.|authorlink=Robert Fricke|title=Ueber eine einfache Gruppe von 504 Operationen|journal=Mathematische Annalen|volume=52|year=1899|pages=321–339|doi=10.1007/BF01476163.

*citation|last=Gofmann|first=R.|title=Weierstrass points on Macbeath's curve|format=In Russian|journal=Vestnik Moskov. Univ. Ser. I Mat. Mekh.|year=1989|issue=5|pages=31–33|volume=104|id=MathSciNet|id=1029778. Translation in "Moscow Univ. Math. Bull." 44 (1989), no. 5, 37–40.

*citation|last=Macbeath|first=A.|authorlink=Alexander M. Macbeath|title=On a curve of genus 7|journal=Proceedings of the London Mathematical Society|volume=15|year=1965|pages=527–542|doi=10.1112/plms/s3-15.1.527.

*citation|last=Vogeler|first=R.|title=On the geometry of Hurwitz surfaces|journal=Florida State University thesis|year=2003.

*citation|last=Wohlfahrt|first=K.|title=Macbeath's curve and the modular group|journal=Glasgow Math. J.|volume=27|year=1985|pages=239–247|id=MathSciNet|id=0819842. Corrigendum, vol. 28, no. 2, 1986, p. 241, MathSciNet|id=0848433.

ee also

* Klein quartic
* First Hurwitz triplet


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Surface de Macbeath — En géométrie hyperbolique, une surface de Macbeath, également appelée courbe de Macbeath ou courbe de Fricke–Macbeath, est une surface de Riemann de genre 7. C est une surface de Hurwitz, c est à dire une surface de Riemann qui possède le nombre… …   Wikipédia en Français

  • Surface de Bolza — Représentation en perspective de l équation y2 = x5 − x dans . En mathématiques, la surface de Bolza (du nom d Oskar Bolza) est une surface de Riemann compacte de g …   Wikipédia en Français

  • Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

  • Bolza surface — Perspective projection of y^2=x^5 x in mathbb C^2.In mathematics, the Bolza surface is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely 48. An affine model for the… …   Wikipedia

  • 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 virtue of… …   Wikipedia

  • Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… …   Wikipedia

  • Projet:Mathématiques/Liste des articles de mathématiques — Cette page n est plus mise à jour depuis l arrêt de DumZiBoT. Pour demander sa remise en service, faire une requête sur WP:RBOT Cette page recense les articles relatifs aux mathématiques, qui sont liés aux portails de mathématiques, géométrie ou… …   Wikipédia en Français

  • Systolic geometry — In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner, and developed by Mikhail Gromov and others, in its arithmetic, ergodic, and topological manifestations.… …   Wikipedia

  • First Hurwitz triplet — In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 admit a unique Hurwitz surface,… …   Wikipedia

  • 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… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”