Nichifor, Alexandra; Palvannan, Bharathwaj

On Free Resolutions of Iwasawa Modules

Doc. Math. 24, 609-662 (2019)
DOI: 10.25537/dm.2019v24.609-662
Communicated by Otmar Venjakob

Summary

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 study of the Dieudonné 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.

Mathematics Subject Classification

11R23, 11R34, 11S25

Keywords/Phrases

(non-commutative) Iwasawa theory, Selmer groups, Galois cohomology

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Affiliation

Nichifor, Alexandra
Department of Mathematics, University of Washington, Seattle WA 98195-4350, USA
Palvannan, Bharathwaj
Department of Mathematics, University of Pennsylvania, Philadelphia PA 19104-6395, USA

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