\(\ell\)-adic embeddings and \(\ell\)-adic Weil numbers
The classical Weil numbers are elements of the algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\) whose images under all embeddings into \(\mathbb{C}\) have absolute value \(1\). These are also referred to as Weil numbers `at infinity' and their connexions with totally real and CM (complex multiplication) fields are reviewed in the first part of this article. The author then defines \(\ell\)-adic analogues of the classical Weil numbers. In addition, he defines the \(\ell\)-adic counterparts to totally real and CM fields, and discusses their basic properties. As an application, some consequences in Iwasawa theory are given. In particular, the conjectures of Gross and Leopoldt come into play and, under certain (strong) hypotheses, a lower bound for for Iwasawa's \(\lambda\)-invariant attached to the \(\ell\)-parts of ideal class groups in a cyclotomic \(\mathbb{Z}_{\ell}\)-extension is given.
scientific article; zbMATH DE number 5543165
scientific article