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  • Print publication year: 2011
  • Online publication date: January 2012

On the MH(G)-conjecture



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.

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