The quotients of the p -adic group ring of a cyclic group of order p

We classify, up to isomorphism, the [Formula: see text]-modules of rank [Formula: see text] (i.e., the quotients of [Formula: see text]) for [Formula: see text] cyclic of order [Formula: see text], where [Formula: see text] is the ring of [Formula: see text]-adic integers. This allows us in particular to determine effectively the quotients of [Formula: see text] which are cohomologically trivial over [Formula: see text]. There are natural zeta functions associated to [Formula: see text] for which we give an explicit formula.