Comparison of two notions of rationality for a dessin d'enfant
Let \(f:{\mathcal C}\to \mathbb{P}^1\) be a ramified covering, where \({\mathcal C}\) is an algebraic curve, defined over an algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\). Let \(\sigma\) be an element of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\), denote by \({}^\sigma f: {}^\sigma{\mathcal C}\to \mathbb{P}^1\) the covering obtained by base change. There are two ways of saying that the covering is ``stable'' under \(\sigma\): 1) there exists \(u_\sigma:{\mathcal C}\simeq {}^\sigma{\mathcal C}\), defined over \(\overline{\mathbb{Q}}\), such that \(f= {}^\sigma fu_\sigma\), 2) there exists \(u_\sigma\) as above and an automorphism \(v_\sigma\) of \(\mathbb{P}^1\), defined over \(\overline{\mathbb{Q}}\), such that \(v_\sigma f= {}^\sigma fu_\sigma\). In both cases, the set of such \(\sigma\) is an open subgroup of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\), then one can define ``fields of definition'' for the covering \(f\). The author compares these two natural notions of rationality, especially when they differ, when the two fields of definition are not the same. Explicit examples are given.
scientific article; zbMATH DE number 1770784