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Equivariant Epsilon Constants, Discriminants and Étale Cohomology

Published online by Cambridge University Press:  23 October 2003

W. Bley
Affiliation:
Institut für Mathematik, Universität Augsburg, Universitätsstrasse 8, D-86135 Augsburg, Germany. E-mail: bley@math.uni-augsburg.de
D. Burns
Affiliation:
Department of Mathematics, King's College London, Strand, London WC2R 2LS. E-mail: david.burns@kcl.ac.uk
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Abstract

Let $L/K$ be a finite Galois extension of number fields. We formulate and study a conjectural equality between an element of the relative algebraic K-group $K_0(\mathbb{Z}[\mathrm{Gal}(L/K)], \mathbb{R})$ which is constructed from the equivariant Artin epsilon constant of $L/K$ and a sum of structural invariants associated to $L/K$. The precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the Tamagawa Number Conjecture of Bloch and Kato (as formulated by Flach and the second-named author) should be compatible with the functional equation of the associated L-function. We show that, more concretely, our conjecture also suggests a completely systematic refinement of the central approach and results of classical Galois module theory. In particular, the evidence for our conjecture that we present here already strongly refines many of the main results of Galois module theory.

Type
Research Article
Copyright
2003 London Mathematical Society

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