On Free Resolutions of Iwasawa Modules

  • Alexandra Nichifor

    Department of Mathematics, University of Washington, Seattle WA 98195-4350, USA
  • Bharathwaj Palvannan

    Department of Mathematics, University of Pennsylvania, Philadelphia PA 19104-6395, USA
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Abstract

Let Λ\Lambda (isomorphic to Zp[[T]]\mathbb{Z}_p[[T]]) denote the usual Iwasawa algebra and GG denote the Galois group of a finite Galois extension L/KL/K of totally real fields. When the non-primitive Iwasawa module over the cyclotomic Zp\mathbb{Z}_p-extension has a free resolution of length one over the group ring Λ[G]\Lambda[G], we prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive pp-adic LL-function (which is an element of a K1K_1-group) in a maximal Λ\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 Q\mathbb{Q} with a cyclic isogeny of degree p2p^2. We relate the characteristic ideal in the ring Λ\Lambda of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over Λ\Lambda, associated to two non-primitive classical Iwasawa modules.

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Alexandra Nichifor, Bharathwaj Palvannan, On Free Resolutions of Iwasawa Modules. Doc. Math. 24 (2019), pp. 609–662

DOI 10.4171/DM/690