On an isomorphism lying behind the class number formula

Vlad Crișan

Let pp be an odd prime such that the Greenberg conjecture holds for the maximal real cyclotomic subfield K1K1 of Q[ζp]Q[ζp]. Let An=(C(Kn))pAn=(C(Kn))p be the pp-part of the class group of KnKn, the nn-th field in the cyclotomic tower, and let E−−nE_n, C−−nC_n be the global and cyclotomic units of KnKn, respectively. We prove that under this premise, there is some n0n0 such that for all mn0m≥n0, the class number formula ∣∣(E−−m/C−−m)p∣∣=|Am||(E_m/C_m)p|=|Am| hides in fact an isomorphism of Λ[Gal(K1/Q)]Λ[Gal(K1/Q)]-modules.

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Crișan, Vlad