Bolza surface

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus ${\displaystyle 2}$ with the highest possible order of the conformal automorphism group in this genus, namely ${\displaystyle GL_{2}(3)}$ of order 48 (the general linear group of ${\displaystyle 2\times 2}$ matrices over the finite field ${\displaystyle \mathbb {F} _{3}}$). Its full automorphism group (including reflections) is the semi-direct product ${\displaystyle GL_{2}(3)\rtimes \mathbb {Z} _{2}}$ of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation

${\displaystyle y^{2}=x^{5}-x}$

in ${\displaystyle \mathbb {C} ^{2}}$. The Bolza surface is the smooth completion of this affine curve. The Bolza curve also arises as a branched double cover of the Riemann sphere with branch points at the six vertices of a regular octahedron inscribed in the sphere. This can be seen from the equation above, because the right-hand side becomes zero or infinite at the six points ${\displaystyle x=0,1,i,-1,-i,\infty }$.

The Bolza surface has attracted the attention of physicists, as it provides a relatively simple model for quantum chaos; in this context, it is usually referred to as the Hadamard–Gutzwiller model.^[1] The spectral theory of the Laplace–Beltrami operator acting on functions on the Bolza surface is of interest to both mathematicians and physicists, since the surface is conjectured to maximize the first positive eigenvalue of the Laplacian among all compact, closed Riemann surfaces of genus ${\displaystyle 2}$ with constant negative curvature. Eigenvectors of the Laplace-Beltrami operator are quantum analogues of periodic orbits, and as a classical analogue of this conjecture, it is known that of all genus ${\displaystyle 2}$ hyperbolic surfaces, the Bolza surface maximizes the length of the shortest closed geodesic, or systole (Schmutz 1993).