On certain birational invariants of the Fermat curves

For an integer m≥3m≥3 let Km=Q(x,y)(xm+ym=1) be the mm-th Fermat field over QQ. We study, in the case of Fermat fields, the groups of integral differentials introduced by Kähler [E. Kähler, Geometria aritmetica, Annali di Mat. 45 (1958)] and Bost [R. Berndt, Arithmetisch ganze Differentiale, Abh. Math. Sem. Univ. Hamburg 47 (1978) 249–270] for arithmetic function fields and compute them for small mm. An essential step of our considerations is the explicit description of the discrete valuation rings with quotient field KmKm which are essentially of finite type and smooth over ZZ.