### A Hasse principle for two dimensional global fields.

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In [6], S. Bloch conjectures a formula for the Artin conductor of the ℓ-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The formula, which we call the conductor formula of Bloch, enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. In this paper, we prove the formula in arbitrary dimension under the assumption that the reduced closed fiber has normal crossings.

Let $X$ be a proper smooth variety over a field $k$ of characteristic $0$ and $Y$ an effective divisor on $X$ with multiplicity. We introduce a generalized Albanese variety Alb$(X,Y)$ of $X$ of modulus $Y$, as higher dimensional analogue of the generalized Jacobian with modulus of Rosenlicht-Serre. Our construction is algebraic. For $k=\u2102$ we give a Hodge theoretic description.

Let G be a compact -adic Lie group, with no element of order , and having a closed normal subgroup H such that G/H is isomorphic to . We prove the existence of a canonical Ore set S of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its...

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