On an isomorphism lying behind the class number formula

Crişan, Vlad



Let p be an odd prime such that the Greenberg conjecture holds for the maximal real cyclotomic subfield \K_1 of \Q[ \zeta_p ]. Let A_n = (\id{C}(\K_n))_p be the p-part of the class group of \K_n, the n-th field in the cyclotomic tower, and let \underline{E}_n, \underline{C}_n be the global and cyclotomic units of \K_n, respectively. We prove that under this premise, there is some n_0 such that for all m \geq n_0, the class number formula \left|\left(\underline{E}_m/\underline{C}_m\right)_p\right|=|A_m| hides in fact an isomorphism of \Lambda[\Gal(\K_1/\Q)]-modules.

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Crişan, Vlad