On Free Resolutions of Iwasawa Modules
| Autor: | Nichifor, Alexandra, Palvannan, Bharathwaj |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Documenta Mathematica. 24:609-662 |
| ISSN: | 1431-0643 1431-0635 |
| DOI: | 10.4171/dm/690 |
| Popis: | 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. Comment: 39 pages, arXiv version 3 - updated acknowledgements, Accepted for publication in Documenta Mathematica |
| Databáze: | OpenAIRE |
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