Let Lambda (isomorphic to 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 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 Dicudonne 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 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.
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Univ Kentucky, Dept Math, 715 Patterson Off Tower, Lexington, KY 40506 USAUniv Kentucky, Dept Math, 715 Patterson Off Tower, Lexington, KY 40506 USA
Morrow, Michael
Nagel, Uwe
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Univ Kentucky, Dept Math, 715 Patterson Off Tower, Lexington, KY 40506 USAUniv Kentucky, Dept Math, 715 Patterson Off Tower, Lexington, KY 40506 USA
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Capital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R ChinaCapital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R China
Lai, King Fai
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Longhi, Ignazio
Tan, Ki-Seng
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Natl Taiwan Univ, Dept Math, Taipei 10764, TaiwanCapital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R China
Tan, Ki-Seng
Trihan, Fabien
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Sophia Univ, Fac Sci & Technol, Dept Informat & Commun Sci, Chiyoda Ku, 4 Yonbancho, Tokyo 1020081, JapanCapital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R China