Bolza surface

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (named after Oskar Bolza), 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 Bolza surface can be obtained as the locus of the equation

$y^2=x^5-x$

in $\mathbb C^2$ . Of all genus 2 hyperbolic surfaces, the Bolza surface has the highest systole. As a hyperelliptic surface, it arises as the ramified double cover of the 2-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.

1 Triangle surface
2 Quaternion algebra
3 See also
4 References

Triangle surface

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. It is interesting to note that 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

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.
Maclachlan, C.; Reid, A. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Math.. 219. New York: Springer. ISBN 0387983864.

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 testing Elliptic curve primality proving

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 for algebraic curves 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

Bolza surface

Contents

Triangle surface

Quaternion algebra

See also

References