Units in number fields and in function fields
Number fields on the one hand and congruence function fields on the other hand share similar properties. For curves over an arbitrary field we have a behavior different from the case of number fields or function fields over a finite field. In this paper, it is considered a question raised by \textit{M. Miyanishi} [J. Algebra 173, No. 1, 144-165 (1995; Zbl 0835.13006)]: If \(\pi: C ^0 \to A^1\) is an affine étale curve over a field \(k\) and all units are constants in the affine coordinate ring \(R = \Gamma(C ^0, {\mathcal O} _{C ^0})\). Does it follow that \(C ^0 \to A^1\) is an isomorphism? If \(\text{char } k > 0\) it is easy to give counter-examples. Miyanishi raised the question with \(k = \mathbb C\) the field of complex numbers. The analogous question in the case of number fields has a positive answer. Before giving the answer to Miyanishi's question, several results and conjectures in number fields and function fields are presented. For example, it is observed that the \(abc\) conjecture is still an open problem in number fields but the analogous problem for function fields is not difficult, as it also happens with the analogs of Fermat's Last Theorem. The author proves that Miyanishi's question has a positive answer for function fields of characteristic zero and \(\deg (\pi) \leq 2\) and also in the case of genus \(\leq 1\). The main result of the paper is that for every \(\deg(\pi) >2\) there is a counterexample defined over \(\mathbb C\).
scientific article; zbMATH DE number 1310243