On the \(p\)-part of the ideal class group of \(\mathbb{Q} (\zeta_ p + \zeta_ p^{-1})\) and Vandiver's conjecture
Let \(p\geq 5\) be prime, and let \(\zeta\) denote a primitive \(p\)-th root of unity. Put \(K= \mathbb{Q} (\zeta)\) and \(G= \text{Gal} (K/Q)\); using the Teichmüller character \(\omega\) we can define idempotents \(e_0, \dots, e_{p-2}\) of \(\mathbb{Z}_p [G]\) corresponding to the powers of \(\omega\) and decompose the \(p\)-Sylow subgroup \(A\) of the ideal class group of \(K\) as well as the \(p\)-Sylow subgroup \(W\) of the unit group of \(K\) modulo the group of circular units into eigenspaces. Results of Mazur, Wiles and Kolyvagin state that, for even \(r\), the cardinalities of \(e_r (A)\) and of \(e_r (W)\) coincide and that, for \(r\) odd, \(\# e_r (A)\) is just the \(p\)-part of the generalized Bernoulli numbers. For fixed even \(r\geq 2\), define \(\beta= \beta_n= \prod_{k=1}^{p-1} (1- \zeta^k )^{k^{(p- 1-r) p^{n-1}}}\) and let \(\ell_n\) denote the largest integer \(\leq n\) such that \(\beta_n\) is a \(p^{\ell_n}\)th power in \(K\). It is known that the sequence \(\ell_n\) eventually stabilizes, and that the cardinalities of \(e_r (A)\) and of \(e_r (W)\) are equal to \(p^{\ell_n}\) for all large enough \(n\). It is these integers \(\ell_n\) which the author studies in this article. In fact, fix \(n\geq 1\) and let \(P_m\) denote the set of all prime ideals in \(K\) which are above the rational primes \(q\equiv 1\bmod p^n\) and which satisfy \(p^{ (q-1)/ p} \equiv \zeta \bmod Q\). Let \(s\) be a primitive root modulo \(Q\) such that \(s^{ (q-1) /p} \equiv \zeta \bmod Q\); then the index \(i(Q)\) defined by \(\beta \equiv s^{i (Q)} \bmod Q\) can be computed in terms of Gaussian periods, and we have \(p^{\ell_n} \leq p^{i (Q)}\) for all \(Q\in P_m\). Moreover, there exists a prime ideal \(Q\in P_m\) such that equality holds. This is the first of many results on the numbers \(i(Q)\) and \(\ell_n\) which are proved in this article but which are too technical to be given here. Suffice it to say that the author succeeds in proving a criterion which allows one to decide whether the component \(e_r (A)\) for even \(r\) is trivial or not (theoretically, at least). The main ingredients of the proofs are Tchebotarev's density theorem (used for reducing the computation of the \(\ell_n\) to local problems), Kummer's complementary reciprocity laws, and the author's characterization of Gaussian periods [Proc. Am. Math. Soc. 124, No. 1, 35-45 (1996)].
scientific article; zbMATH DE number 838733