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Stable reduction of Fermat curves and Jacobi sum Hecke characters. (English) Zbl 0654.12003

The authors exhibit precise formulae for the local components \(\alpha_{{\mathfrak p}}\) (and conductors) of Jacobi sum characters (for a prime ideal \({\mathfrak p}\nmid 2\) in the cyclotomic field \(\mathbb Q(\exp (2\pi i/m)))\). The method involves the action of the Galois group on the stable reduction of an appropriate Fermat curve. Using the Hasse- Davenport relation for Gauß sums, a general formula for \(\alpha_{{\mathfrak p}}\), when \(3| m\) and \(3\in {\mathfrak p}\), is obtained as a consequence.

MSC:

11R18 Cyclotomic extensions
11L10 Jacobsthal and Brewer sums; other complete character sums
14H25 Arithmetic ground fields for curves
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