Mathematics – Complex Variables
Scientific paper
2012-03-28
Mathematics
Complex Variables
Scientific paper
The well known Hurwitz upper bound states that a closed Riemann surface $S$ of genus $g \geq 2$ has at most $84(g-1)$ conformal automorphisms. If it has exactly $84(g-1)$ conformal automorphisms, then it is called a Hurwitz curve. The first two genera for which there are Hurwitz's curves are $g \in \{3,7\}$. In both situations there is exactly one such curve up to conformal equivalence, in particular, in both cases the field of moduli is ${\mathbb Q}$. As these two curves are quasiplatonic curves, they are definable over ${\mathbb Q}$. The Hurwitz's curve of genus $g=3$ is given by Klein's quartic $x^3y+y^3z+z^3x=0$. The Hurwitz's curve of genus $g=7$ is known as Fricke-Macbeath's curve and equations over ${\mathbb Q}(\rho)$, where $\rho=e^{2 \pi i/7}$, are known due to Macbeath. Unfortunately, no known explicit equations of ${\mathbb Q}$ are given in the literature for this curve. In this paper we first provide an explicit model $Z_{2}$ of Fricke-Macbeath's curve in the degree two extension ${\mathbb Q}(\sqrt{-7})$ and an explicit isomorphism $L:X \to Z_{2}$ (and its inverse $R:Z_{2} \to X$). Using that explicit model we construct another explicit isomorphism $R_{2}:Z_{2} \to W$, where $W$ is some algebraic curve which can be defined by polynomials over ${\mathbb Q}$. Unfortunately, the equations for $W$ are quite long to write down, but everything is explained in order to perform the computations in a computer.
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