On an isomorphism lying behind the class number formula

Crişan, Vlad

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carpathian_33_1_043_048_abstract

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carpathian_2017_33_1_43_48

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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.