Die stabile Reduktion der Fermatkurve über einem Zahlkörper. (The stable reduction of the Fermat curve over a number field)
Let p be an odd prime and F the Fermat curve of degree p, defined by \(x^ p+y^ p=1\) over \({\mathbb{Q}}\). Although F has bad reduction at the prime (p), according to the stable reduction theorem (Grothendieck, Mumford) we can have stable reduction of F over some number field \(K/{\mathbb{Q}}\). The author determines this in rather elementary way. For \(p=3\), \textit{I. R. Shafarevich} has observed [see Transl., II. Ser., Am. Math. Soc. 31, 25-39 (1963); translation from Proc. internat. Congr. Math., Stockholm 1962, 163-176 (1963; Zbl 0126.069)] that F has everywhere good reduction over \({\mathbb{Q}}(^ 3\sqrt{2},^ 4\sqrt{-3})\). In the case of \(p>3\) the stable reduction of F consists of one rational curve and \(p-2\) times Artin-Schreier hyperelliptic curves: \(y^ 2=x^ p-x.\) Coleman and McCallum have already obtained a stable model of the Fermat quotient curves: \(y^ m=(-1)^ cx^ a(1-x)^ b.\)
scientific article; zbMATH DE number 3937327
scientific article