Stickelberger subideals for a prime modulus related to Kummer-type congruences
Let \(p\) be a prime \(\geq 5\) and let \(G\) denote a cyclic group of order \(p-1\). For a set \(\Gamma\) of different integers in the interval \([2,p-1]\), the author introduces a subideal \(\mathcal I_\Gamma (p)\) of the Stickelberger ideal \(\mathcal I(p)\) in the group ring \((\mathbb{Z}/ p\mathbb{Z})[G]\). He finds various bases (over \(\mathbb{Z}/p\mathbb{Z}\)) of this subideal. They are closely related to the following ``Kummer-type'' system of congruences: \[ \phi _{p-1}(t) \equiv 0,\quad B_{2m}^{(\Gamma)} \phi _{p-2m}(t) \equiv 0 \pmod p, \qquad (m=1,\dots,(p-3)/2). \] Here \(\phi _k(x)\) is the Mirimanoff polynomial and \(B_k^{(\Gamma)} = \prod _{N \in \Gamma} (N^k-1)B_k/k\), where \(B_k\) denotes the \(k\)-th Bernoulli number. Through an index formula the author establishes a relationship between his subideal and the relative class number of the \(p\)-th cyclotomic field. Moreover he shows that dim\(\mathcal I_\Gamma (p)\) equals \((p-1)/2\) minus the number of \(p\)-divisible \(B_{2m}^{(\Gamma)}\). According to the author, many of the ideas are adaptations or generalizations of those by \textit{L. Skula} [see, e.g., J. Number Theory 58, No. 1, 173-195 (1996; Zbl 0861.11063)]. The original Kummer system of congruences appeared in the study of Fermat's equation.
scientific article; zbMATH DE number 1109647
scientific article