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THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS

Part of: Modules, bimodules and ideals Rings and algebras with additional structure

Published online by Cambridge University Press:  02 October 2015

XIN WAN
XIN WAN*
Affiliation:
Mathematics Department, Columbia University, New York, 10027, USA; [email protected]

Abstract

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Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing $p$, if the central critical $L$-value is zero then the $p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

MSC classification

Primary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures
Secondary: 16D50: Injective modules, self-injective rings
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e18
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2015

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