GLOBAL UNITS MODULO ELLIPTIC UNITS AND IDEAL CLASS GROUPS

Abstract

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two distinct primes 𝔭 and [Formula: see text]. Let k∞be the unique ℤp-extension of k which is unramified outside of 𝔭, and let K∞be a finite extension of k∞, abelian over k. Following closely the ideas of Belliard in [1], we prove that in K∞, the projective limit of the p-class group and the projective limit of units modulo elliptic units share the same μ-invariant and the same λ-invariant. We deduce that a version of the classical main conjecture, which is known to be true for p ∉ {2, 3}, holds also for p ∈ {2, 3} once we neglect the μ-invariants.