On certain birational invariants of the Fermat curves (Q878660)

Let \(K_{m} = {\mathbb Q}(x, y),\) \(x^{m}+y^{m}=1\) (\(m > 2)\) be the field of rational functions of the Fermat curve over \({\mathbb Q}\), and let \(V\) be the set of all discrete valuation rings \((R, m_{R})\) with \(Q(R)=K_{m}\) which are essentially of finite type over \({\mathbb Z}\). Moreover, let \(V_{s}=\{R\in V: R \text{ is smooth over}\, {\mathbb Z}\}\). The authors study, in the case of Fermat fields, the groups of integral differentials introduced by Kähler and Bost for arithmetic function fields. They derive some properties of these birational invariants of the Fermat curves and compute them for small \(m;\) describe the elements of \(V_{s}\) and their modules of differentials explicitly, which is done in section 3. This section is based on the explicit description of the discrete valuation rings of the rational functions field \(\mathbb Q(x)\), which are essentially of finite type and smooth over \(\mathbb Z\), and of their modules of differentials.