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- W4313854790 abstract "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. When the non-primitive Iwasawa module over the cyclotomic $mathbb{Z}_p$-extension has a free resolution of length one over the group ring $Lambda[G]$, we prove that 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. This integrality result involves a careful study of the Dieudonn'e determinant. Using a cohomolgoical criterion of Greenberg, we also deduce the precise conditions under which the non-primitive Iwasawa module has a free resolution of length one. As one application of the last result, 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." @default.
- W4313854790 created "2023-01-10" @default.
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- W4313854790 date "2017-07-05" @default.
- W4313854790 modified "2023-10-14" @default.
- W4313854790 title "On free resolutions of Iwasawa modules" @default.
- W4313854790 doi "https://doi.org/10.48550/arxiv.1707.01485" @default.
- W4313854790 hasPublicationYear "2017" @default.
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