We describe an algorithm for finding the coefficients of F (X) modulo powers of p, where p = 2 is a prime number and F (X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A3 (up to field discriminant <10 7 ), and also tabulate the class number of K(e 2πi/p ) for p = 5 and p = 7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K.
We completely describe K 1 (Z p [[G ∞ ]]) and its localisations by using an infinite family of p-adic congruences, where G ∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G ∞ ) = 2, and of the first named author and Lloyd Peters when G ∞ ∼ = Z × p Z d p with a scalar action of Z × p . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory.
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