Note sur les diviseurs \'el\'ementaires du r\'egulateur d'Iwasawa

The article revisits a result of Antonio Lei, David Loeffler, and Sarah Livia Zerbes concerning the structure of the image by the Iwasawa regulator map of the Iwasawa module associated with a semi-stable $p$-adic representation on an unramified finite extension of $Q_p$ and give a direct proof based on the results of “Repr'esentations $p$-adiques et normes universelles: I. le cas cristallin” in the crystalline case and of “Th'eorie d’Iwasawa des repr'esentations $p$-adiques semi-stables” in the semi-stable case.