On free resolutions of Iwasawa modules

Alexandra Nichifor, Bharathwaj Palvannan

Published 2017-07-05Version 1

Let $\Lambda$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of totally real fields. The main theorems in this article describe the precise conditions under which non-primitive Iwasawa modules, over the cyclotomic $\mathbb{Z}_p$-extension, have a free resolution of length one over the group ring $\Lambda[G]$. As one application, under these conditions of the main theorems, the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive $p$-adic $L$-function (which is an element of a $K_1$-group) in a maximal $\Lambda$-order. As another application, we consider an elliptic curve over $\mathbb{Q}$ with a cyclic isogeny of degree $p^2$. We relate the characteristic ideal, in the ring $\Lambda$, of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over $\Lambda$, associated to two non-primitive classical Iwasawa modules.