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On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results


Authors: Henri Johnston and Andreas Nickel
Journal: Trans. Amer. Math. Soc. 368 (2016), 6539-6574
MSC (2010): Primary 11R42, 19F27
Published electronically: August 20, 2015
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Abstract: Let $ L/K$ be a finite Galois extension of number fields with Galois group $ G$. Let $ p$ be a prime and let $ r \leq 0$ be an integer. By examining the structure of the $ p$-adic group ring $ \mathbb{Z}_{p}[G]$, we prove many new cases of the $ p$-part of the equivariant Tamagawa number conjecture (ETNC) for the pair $ (h^{0}(\mathrm {Spec}(L))(r),\mathbb{Z}[G])$. The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic $ K$-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a large class of interesting extensions, including cases in which the full ETNC is not known. Similarly, we construct annihilators of higher dimensional algebraic $ K$-groups of the ring of integers in $ L$.


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Additional Information

Henri Johnston
Affiliation: Department of Mathematics, University of Exeter, Harrison Building, Exeter, EX4 4QF, United Kingdom
Email: H.Johnston@exeter.ac.uk

Andreas Nickel
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, Universitätsstr. 25, 33501 Bielefeld, Germany
Email: anickel3@math.uni-bielefeld.de

DOI: https://doi.org/10.1090/tran/6453
Keywords: Tamagawa number, algebraic $K$-groups, annihilators, class groups
Received by editor(s): March 6, 2014
Received by editor(s) in revised form: August 26, 2014
Published electronically: August 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society