Mathematics – Complex Variables
Scientific paper
2012-03-28
Mathematics
Complex Variables
Scientific paper
It is well known that every closed Riemann surfaces of genus zero or one can be defined over their fields of moduli and that such fields are easily computable. For genus at least two, both, the computation of the field of moduli and to determine if it is a field of definition, are in general very difficult tasks. In this paper we consider the family of closed Riemann surfaces of genus three admitting as a group of conformal automorphisms the symmetric group in four letters ${\mathfrak S}_{4}$. Up to conformal equivalence, there is only one such Riemann surface which is hyperelliptic and it is known for it a hyperelliptic curve over ${\mathbb Q}$. For the non-hyperelliptic ones, all of them, with the exception of two classes (Fermat's quartic and Klein's quartic), have ${\mathfrak S}_{4}$ as full group of conformal automorphisms. Both, Fermat's quartic and Klein's quartic are known to be definable over ${\mathbb Q}$ and such explicit curves are well known. There is also known in the literature a family of non-singular quartics, known as the KFT family. Each member of that family represents a closed Riemann surface of genus three with ${\mathfrak S}_{4}$ as a group of conformal automorphisms and, conversely, every such type of closed Riemann surfaces are represented by a member of the KFT family. We compute the field of moduli for each of these curves and we obtain that, with the exception of Fermat's curve and Klein's quartic, they are all already defined in the KFT family over their fields of moduli.
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