On the mu and lambda invariants of the logarithmic class group

Abstract

Let ℓ be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Zℓ-extension K∞ of a number field K, we show that there exist integers μ˜, λ˜ and ν˜ such that the exponent e˜n of the order ℓe˜n of the logarithmic class group Cℓ˜n for the n-th layer Kn of K∞ is given by e˜n=μ˜ℓn+λ˜n+ν˜, for n big enough. We show some relations between the classical invariants μ and λ, and their logarithmic counterparts μ˜ and λ˜ for some class of Zℓ-extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.