Abstract
Let p be an odd prime and F the Fermat curve of degree p, defined by xp+yp=1 over ℚ. Although the curve F has bad reduction at the prime (p), the stable reduction theorem assures that over some number field K/ℚ we can get stable reduction of the curve F at the primes lying above p. We have determined it in this paper. See Abb.1.
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ARWIN, A.: Über die Lösung der Kongruenz (λ+1)p−λp−1≡0 (mod p2). Acta math.42, 173–190 (1919)
DELIGNE, P., MUMFORD, D.: The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S.36, 75–109 (1969)
FADDEEV, D.K.: Über die Invarianten der Divisorklassen der Kurve xk(1−x)=yℓ über ℓ-adischen Kreisteilungskörpern (Russisch). Trudy Mat. Inst. Steklov64, 284–293 (1961)
FALTINGS, G.: Calculus on arithmetic surfaces. Ann. of Math.119, 387–424 (1984)
GROTHENDIECK, A.: Modéles de Néron et monodromie. in SGA 7I, Lect. Notes in Math.288 Berlin-Heidelberg-New York: Springer-Verlag 1972
LANG, S.: Complex Multiplication. New York-Berlin-Heidelberg-Tokyo: Springer-Verlag 1983
LANGE, H.: Kurven mit rationaler Abbildung. J. reine angew. Math.295, 80–115 (1977)
MAZUR, B.: The cohomology of the Fermat group scheme. Appendix to GROSS, B.H.: Arithmetic of elliptic Curves with complex multiplication. Lect. Notes in Math.776 Berlin-Heidelberg-New York: Springer-Verlag 1980
McCALLUM, W.: The degenerate fiber of the Fermat curve. in “Number Theory related to Fermat's Last Theorem”. Progress in Math.26, 57–70 Boston-Basel-Stuttgart: Birkhäuser 1982
OORT, P., SEKIGUCHI, T.: On the Deformation of Artin-Schreier to Kummer. Preprint Nr.369, University of Utrecht 1985
RAYNAUD, M.: Spéialisation du Foncteur de Picard. Publ. Math. I.H.E.S.38, 27–76 (1970)
SCHMIDT, C.-G.: Arithmetik Abelscher Varietäten mit komplexer Multiplikation. Lect. Notes in Math.1082 Berlin-Heidelberg-New York-Tokyo: Springer-Verlag 1984
SERRE, J.-P., TATE, J.: Good reduction of abelian varieties. Ann. of Math.88, 492–517 (1968)
SHIODA, T.: What is known about the Hodge conjecture? in “Algebraic Varieties and Analytic Varieties”. Advanced Sudies in Pure Math.1, Kinokuniya-North Holland 1983
SULLIVAN, P.J.: p-torsion in the class group of curves with too many automorphismus. Arch. d. Math.26, 253–261 (1975)
SZPIRO, L.: Séminaire sur les pinceaux de courbes de genre au moins deux. Astérisque86, Soc. Math. de France 1981
VIEHWEG, E.: Invarianten der degenerierten Fasern in lokalen Famillien von Kurven. J. reine angew. Math.293/294, 284–308 (1977)
WILLIAMSON, S.: Ramification theory for extensions of degree p. Nagoya Math. J.41, 149–168 (1971)
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Maeda, H. Die stabile Reduktion der Fermatkurve über einem Zahlkörper. Manuscripta Math 56, 333–342 (1986). https://doi.org/10.1007/BF01180772
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DOI: https://doi.org/10.1007/BF01180772