On Free Resolutions of Iwasawa Modules

Alexandra Nichifor ; Bharathwaj Palvannan

Documenta mathematica, Tome 24 (2019), pp. 609-662

Résumé

Let Λ (isomorphic to Zp​[[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 Zp​-extension has a free resolution of length one over the group ring Λ[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 K1​-group) in a maximal Λ-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 with a cyclic isogeny of degree p2. We relate the characteristic ideal in the ring Λ of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over Λ, associated to two non-primitive classical Iwasawa modules.

Détail

DOI : 10.4171/dm/690

Classification : 11R23, 11R34, 11S25
Mots-clés : Selmer groups, Galois cohomology, (non-commutative) Iwasawa theory