Bolza surface

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL₂(3) of order 48. An affine model for the Bolza surface can be obtained as the locus of the equation

y^{2}=x^{5}-x

in ${\mathbb C}^{2}$ . The Bolza surface is the smooth completion of the affine curve. Of all genus 2 hyperbolic surfaces, the Bolza surface has the highest systole. As a hyperelliptic Riemann surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six vertices of a regular octahedron inscribed in the sphere, as can be readily seen from the equation above.

Triangle surface

The tiling of the Bolza surface by reflection domains is a quotient of the order-3 bisected octagonal tiling.

The Bolza surface is a (2,3,8) triangle surface – see Schwarz triangle. More specifically, the Fuchsian group defining the Bolza surface is a subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles ${\tfrac {\pi }{2}},{\tfrac {\pi }{3}},{\tfrac {\pi }{8}}$ . More specifically, it is a subgroup of the index-two subgroup of the group of reflections, which consists of products of an even number of reflections, which has an abstract presentation in terms of generators $s_{2},s_{3},s_{8}$ and relations $s_{2}{}^{2}=s_{3}{}^{3}=s_{8}{}^{8}=1$ as well as $s_{2}s_{3}=s_{8}$ . The Fuchsian group defining the Bolza surface is also a subgroup of the (3,3,4) triangle group, which is a subgroup of index 2 in the (2,3,8) triangle group. The (2,3,8) group does not have a realisation in terms of a quaternion algebra, but the (3,3,4) group does.

Quaternion algebra

Following MacLachlan and Reid, the quaternion algebra can be taken to be the algebra over $\mathbb {Q} ({\sqrt {2}})$ generated as an associative algebra by generators i,j and relations

i^{2}=-3,\;j^{2}={\sqrt {2}},\;ij=-ji,

with an appropriate choice of an order.

References

Bolza, Oskar (1887), "On Binary Sextics with Linear Transformations into Themselves", American Journal of Mathematics, 10 (1): 47–70, JSTOR 2369402, doi:10.2307/2369402
Katz, M.; Sabourau, S. (2006). "An optimal systolic inequality for CAT(0) metrics in genus two". Pacific J. Math. 227 (1): 95–107. arXiv:math.DG/0501017 . doi:10.2140/pjm.2006.227.95.
Maclachlan, C.; Reid, A. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Math. 219. New York: Springer. ISBN 0-387-98386-4.

Topics in algebraic curves

Rational curves

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Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

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Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem
Singularities	Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

Systolic geometry
1-systoles of surfaces	Loewner's torus inequality Pu's inequality Filling area conjecture Bolza surface Systoles of surfaces Eisenstein integers
1-systoles of manifolds	Gromov's systolic inequality for essential manifolds Essential manifold Filling radius Hermite constant
Higher systoles	Gromov's inequality for complex projective space Systolic freedom Systolic category

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Bolza surface

Triangle surface

Quaternion algebra

See also

References