Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



An Equivariant Main Conjecture in Iwasawa theory and applications

Authors: Cornelius Greither and Cristian D. Popescu
Journal: J. Algebraic Geom. 24 (2015), 629-692
Published electronically: July 6, 2015
MathSciNet review: 3383600
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Abstract: We construct a new class of Iwasawa modules, which are the number field analogues of the $ p$-adic realizations of the Picard $ 1$-motives constructed by Deligne and studied extensively from a Galois module structure point of view in our previous works. We prove that the new Iwasawa modules are of projective dimension $ 1$ over the appropriate profinite group rings. In the abelian case, we prove an Equivariant Main Conjecture, identifying the first Fitting ideal of the Iwasawa module in question over the appropriate profinite group ring with the principal ideal generated by a certain equivariant $ p$-adic $ L$-function. This is an integral, equivariant refinement of the classical Main Conjecture over totally real number fields proved by Wiles. Finally, we use these results and Iwasawa co-descent to prove refinements of the (imprimitive) Brumer-Stark Conjecture and the Coates-Sinnott Conjecture, away from their $ 2$-primary components, in the most general number field setting. All of the above is achieved under the assumption that the relevant prime $ p$ is odd and that the appropriate classical Iwasawa $ \mu $-invariants vanish (as conjectured by Iwasawa).

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Cornelius Greither
Affiliation: Institut für Theoretische Informatik und Mathematik, Universität der Bundeswehr, München, 85577 Neubiberg, Germany

Cristian D. Popescu
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112

Received by editor(s): August 13, 2012
Received by editor(s) in revised form: January 30, 2013
Published electronically: July 6, 2015
Additional Notes: The second author was partially supported by NSF Grants DMS-0600905 and DMS-0901447
Article copyright: © Copyright 2015 University Press, Inc.

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