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 Z ℓ-extension K∞ of a number field K, we show that there exist integers µ, λ and ν such that the exponentẽn of the order ℓẽ n of the logarithmic class group Cℓn for the n-th layer Kn of K∞ is given byẽn = µℓ n + λn + 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 Z ℓ-extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case. * K ≃ Gal(K lc /K) is a noetherian Z ℓ-module, and assuming the Gross-Kuz'min Conjecture it has rank 1. This yields the following result.

Topological behaviour of logarithmic invariants

“…Assume that K ∞ splits finitely. By Theorem 4.8 in [9] the classical µ invariant coincides with the logarithmic µ invariant. By Theorem 3.3 in [1], we know that the classical µ invariants are bounded in ∆(K).…”

“…In order to prove the torsion property suppose d ≥ 2, otherwise we know that the result is just Proposition 3.1 in [9] in the non-cyclotomic case and for the cyclotomic Z ℓ -extension this follows from Section 2.5.…”

“…In this work we study these results from the logarithmic arithmetic point of view. That is, our objects will be logarithmic as in [5,9].…”

“…V]. For information on logarithmic arithmetic we invite the reader to consult [5,9]. At the end of the section we recall what we know about the cyclotomic Z ℓ -extension and we introduce the topologies which will be used in this work.…”