On an isomorphism lying behind the class number formula

Abstract

Let p be an odd prime such that the Greenberg conjecture holds for the maximal real cyclotomic subfield K1 of Q[ζp]. Let An = (C(Kn))p be the p-part of the class group of Kn, the n-th field in the cyclotomic tower, and let En, Cn be the global and cyclotomic units of Kn, respectively. We prove that under this premise, there is some n0 such that for all m ≥ n0, the class number formula (Em/Cm)p = |Am| hides in fact an isomorphism of Λ[Gal(K1/Q)]-modules.