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The group of automorphisms of the Fermat curve. (English) Zbl 0853.14015
Summary: In his paper [cf. “Œuvres scientifiques”, Collected Papers, Vol. III (1964-1978), 329-342 (1979; Zbl 0424.01029)], A. Weil asserted (without proof) that the automorphism group of the Fermat hypersurface of exponent $$N$$ and dimension $$r - 1$$ over an algebraically closed field of characteristic prime to $$N$$ is the semidirect product of the symmetric group on $$r + 1$$ letters and the direct sum of $$r$$ copies of the cyclic group of order $$N$$. It turns out that the assertion is false in positive characteristic. In this paper, we present a proof of Weil’s assertion for the case of the Fermat curves $$(r = 2)$$ in characteristic 0.

##### MSC:
 14H30 Coverings of curves, fundamental group 14E07 Birational automorphisms, Cremona group and generalizations
##### Keywords:
automorphisms group of the Fermat curve
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