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Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

Published online by Cambridge University Press:  27 April 2011

ANDREAS NICKEL
ANDREAS NICKEL*
Affiliation:
Universität Regensburg, Fakultät für Mathematik, Universitätsstr. 31, 93053 Regensburg, Germany. e-mail: andreas.nickel@mathematik.uni-regensburg.de
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Abstract

Let L/K be a finite Galois extension of number fields with Galois group G. We use leading terms of Artin L-series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K-groups of the ring of integers in L. For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well-known Coates–Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen–Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h0(Spec(L))(r), where r is a strictly negative integer. In particular, this implies the ETNC for the pair (h0(Spec(L))(r), ), where L is totally real, r < 0 is odd and is a maximal order containing ℤ[]G, and will also provide some evidence for our conjecture.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 1 , July 2011 , pp. 1 - 22
Copyright
Copyright © Cambridge Philosophical Society 2011

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