Vandiver revisited
Fermat's last theorem, the assertion that there are no non-zero integer solutions \(x, y, z, n>2\) to \(x^n+ y^n= z^n\), was recently proved by \textit{A. Wiles} [Ann. Math., II. Ser. 141, 443-551 (1995; Zbl 0823.11029)], after several centuries of work by many mathematicians. Vandiver, during the first half of this century, tried to develop the theory of cyclotomic fields to a state that one could finally prove Fermat's last theorem -- indeed Kummer had already shown that if there were to be a solution with \(n=p\) prime then the \(p\)th cyclotomic field would have to have many very strange and unlikely properties. At one point \textit{H. S. Vandiver} mistakenly claimed [Bull. Am. Math. Soc. 40, 118-126 (1934; Zbl 0009.00701)] that he could prove that if a prime \(p\) does not divide the class number, \(h^+\), of \(\mathbb{Q} (\zeta_p+ \zeta_p^{-1})\) (where \(\zeta_p= e^{2i\pi/ p})\) then any solution to \(x^p+ y^p= z^p\) has \(xyz\) divisible by \(p\). It is conjectured (apparently by both Kummer and Vandiver independently) that \(p\) never does divide this class number. Since Vandiver's paper contained many interesting ideas there have been several attempts to resurrect his proof by patching up his mistakes. The paper under review is born from such an attempt. Fix a prime \(p\), and try to find a positive integer \(k\) such that \(x^n+ y^n= z^n\) has no integer solutions where \(p\) does not divide \(x\), \(y\), \(z\), and where \(n= p^k\). \textit{L. C. Washington} [J. Reine Angew. Math. 289, 115-117 (1977; Zbl 0337.10010)] showed that this is so provided \(p^{k+[ \sqrt {p} ]-3}\) does not divide \(h/ h^+\), where \(h\) is the class number of \(\mathbb{Q} (\zeta_p)\). The \(p\)-Sylow subgroup, \(C_p\), of the class group of \(\mathbb{Q} (\zeta_p)\), can be decomposed into a direct sum of eigenspaces \(C_p^{(j)}\). The Herbrand-Ribet theorem states \(C_p^{(j)}\) is non-trivial if and only if \(p\) divides the Bernoulli number \(B_{p-j}\). It was Cauchy who showed that if there is a solution to \(x^p+ y^p= z^p\) with \((p, xzy)= 1\) then \(p\) divides \(B_{p-3}\), and thus we know that \(C_p^{(3)}\) is non-trivial. Moreover, \textit{M. Kurihara} (Prop. 4.5 in [Compos. Math. 81, 223-236 (1992; Zbl 0747.11055)]) showed that \(C_p^{(3)}\) must then have an element of order \(p^2\). Herein the author generalizes this result by showing that the integer \(k\) will work above provided \(C_p^{(3)}\) has \(\leq p^k\) elements. The proof uses in an essential way ideas of Vandiver, ideas from Mazur and Wiles' proof of the `Main Conjecture', as well as the deep results of Kurihara that \(C_p^{(3)}\) is cyclic and \(C_p^{(p- 3)}\) is trivial.
scientific article; zbMATH DE number 874234