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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