Twisted Fermat curves over totally real fields

Abstract

Let p be a prime number, F a totally real field such that [F(mup): F]=2 and [F:Q] is odd. For delta ∈ Ftimes, let [delta] denote its class in Ftimes/Ftimes p. In this paper, we show Main Theorem. There are infinitely many classes [delta]∈ Ftimes/Ftimes p such that the twisted affine Fermat curves Wdelta: Xp+Yp=delta have no F-rational points.

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