On the <span class='italic'>M<span class='sub'>H</span>(G)</span>-conjecture

doi:10.1017/CBO9780511984440.006

On the MH(G)-conjecture

Published online by Cambridge University Press: 05 January 2012

J. Coates and

Edited by

Mohamed Saïdi and

J. Coates: Affiliation:
University of Cambridge
R. Sujatha: Affiliation:
Tata Institute of Fundamental Research
John Coates: Affiliation:
University of Cambridge
Minhyong Kim: Affiliation:
University College London
Florian Pop: Affiliation:
University of Pennsylvania
Mohamed Saïdi: Affiliation:
University of Exeter
Peter Schneider: Affiliation:
Universität Münster

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Introduction

Let p be any prime number, and let G be a compact p-adic Lie group with a closed normal subgroup H such that G/H is isomorphic to the additive subgroup of p-adic integers ℤp. Write ∧(G) (respectively, ∧(H)) for the Iwasawa algebra of G (respectively, H) with coefficients in ℤp. As was shown in [5], there exists an Ore set in ∧(G) which enables one to define a characteristic element, with all the desirable properties, for a special class of torsion ∧(G)-modules, namely those finitely generated left ∧(G)-modules W such that W/W(p) is finitely generated over ∧(H); here W(p) denotes the p-primary submodule of W. This simple piece of pure algebra leads to a class of deep arithmetic problems, which will be the main concern of this paper. We shall loosely call these problems the MH(G)-conjectures, and it should be stressed that their validity is essential even for the formulation of the main conjectures of non-commutative Iwasawa theory.

Let F be a finite extension of ℚ, and F∞ a Galois extension of F satisying (i) G = Gal(F∞/F) is a p-adic Lie group, (ii) F∞/F is unramified outside a finite set of primes of F, and (iii) F∞ contains the cyclotomic ℤp-extension of F, which we denote by Fcyc.

Type: Chapter
Information: Non-abelian Fundamental Groups and Iwasawa Theory , pp. 132 - 161

DOI: https://doi.org/10.1017/CBO9780511984440.006 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2011

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On the MH(G)-conjecture

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