On the Leopoldt conjecture on the p-adic regulators
Leopoldt's conjecture predicting the non-vanishing of the p-adic regulator \(R_ p(k)\) of an algebraic number field k has been proven by Ax and Brumer in case k is an abelian extension of \({\mathbb{Q}}\) or of any imaginary quadratic number field, using an analytic method of Baker. In the present paper the author investigates the problem how to conclude purely algebraically the non-vanishing of \(R_ p(K)\) for a cyclic extension K/k of degree p under the assumption \(R_ p(k)\neq 0\). For \(k={\mathbb{Q}}\) or k imaginary quadratic this is carried out successfully and moreover an explicite lower bound for \(| R_ p(K)|_ p\) is given for \(k={\mathbb{Q}}\). The conclusion is also proven in a more general situation under certain strong conditions on k and on the ramification of K/k. Furthermore the Galois group \(G_ k(p)\) of the maximal p-ramified abelian p-extension of k is taken into consideration. For a Galois extension K/k of p-power degree a condition is worked out which from \(R_ p(k)\neq 0\) and \(G_ k(p)\) torsion free concludes \(R_ p(K)\neq 0\) and \(G_ K(p)\) torsion free.
scientific article; zbMATH DE number 4006375