Abstract
For an integer m > 3 let K-m = Q(x, y) (x(m) + y(m) = 1) be the m-th Fermat field over Q. We study, in the case of Fermat fields, the groups of integral differentials introduced by Kahler [E. Kahler, 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 ...
Abstract
For an integer m > 3 let K-m = Q(x, y) (x(m) + y(m) = 1) be the m-th Fermat field over Q. We study, in the case of Fermat fields, the groups of integral differentials introduced by Kahler [E. Kahler, 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 m. An essential step of our considerations is the explicit description of the discrete valuation rings with quotient field Km which are essentially of finite type and smooth over Z. (c) 2006 Elsevier B.V. All rights reserved.
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