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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
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