Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals
Let \(S\) be a compact Riemann surface. Then \(S\) can be described as a non-singular complex projective algebraic curve \(C\) inside some projective space by homogeneous polynomials \(P_{1},\dots ,P_{r}\) with coefficients in a subfield \(K\) of \({\mathbb C}\) called a field of definition of \(S\). For a given \(\sigma \in \text{Aut} ({\mathbb C})\), we can construct new polynomials \(P_{1}^{\sigma} ,\dots ,P_{r}^{\sigma}\) by applying \(\sigma\) to the coefficients of the \(P_{i}\). Let \(C^{\sigma}\) denote the algebraic curve defined by \(P_{1}^{\sigma} ,\dots ,P_{r}^{\sigma}\) and let \(S^{\sigma}\) denote the corresponding compact Riemann surface. Let \(G\) denote the subgroup of \(\text{Aut} ({\mathbb C} )\) consisting of all \(\sigma \in \text{Aut} ({\mathbb C} )\) such that \(S\) and \(S^{\sigma}\) are conformally equivalent. Then we call the fixed field of \(G\) the moduli field of \(S\). Clearly the moduli field is a subfield of any field of definition of \(S\). The moduli field however need not be a field of definition, and there are specific examples of families of hyperelliptic surfaces where this is not the case, see for example [\textit{Y. Fuertes} and \textit{G. González-Diez}, Arch. Math. 86, No. 5, 398--408 (2006; Zbl 1095.14028)]. However, no previous examples exist for surfaces which are not hyperelliptic. In the paper under review, the author constructs an explicit example of a non-hyperelliptic family of compact Riemann surfaces where the field of moduli is not a field of definition. The results of the author rely on two main results. First, a compact Riemann surface \(S\) has moduli field contained in \({\mathbb R}\) if and only if \(S\) admits an anticonformal automorphism. Second, \(S\) has a field of definition contained in \({\mathbb R}\) if and only if \(S\) admits an anticonformal involution of order \(2\). Using these two results, it suffices to exhibit an explicit family of non-hyperelliptic compact Riemann surfaces which admit some anticonformal automorphism, but do not admit an anticonformal automorphism of order \(2\). To do this, the author first provides an explicit set of polynomials \(P_{1},\dots ,P_{r}\) for a family of algebraic curves of genus \(17\). By using these explicit equations, the author is able to prove the existence of a group of automorphism \(H={\mathbb Z}_{2}^{5}\) such that the quotient map \(\pi : S\rightarrow S/H\) is branched over \(6\) points which depend upon the coefficients of the \(P_{i}\), and with the property that \(S/H\) has genus \(0\). The author then uses these equations to construct an explicit anticonformal automorphism of order \(4\) whose square lies in \(H\) (and hence is conformal). It follows that the moduli field is contained in \({\mathbb R}\). Next, the author shows that there is no anticonformal automorphism of order \(2\). Specifically, the existence of such an automorphism guarantees the existence of a Mobius transformation which preserves the branch locus of the map \(\pi : S\rightarrow S/H\), and since the branching locus is explicitly know (as it depends on the \(P_{i}\)), it can be shown that no such Mobius transformation exists. The paper is well written and ideal for both experts and non-experts in the area.
scientific article; zbMATH DE number 5636258
scientific article