On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results
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- by Henri Johnston and Andreas Nickel PDF
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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 $L$.References
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Additional Information
- Henri Johnston
- Affiliation: Department of Mathematics, University of Exeter, Harrison Building, Exeter, EX4 4QF, United Kingdom
- MR Author ID: 776746
- ORCID: 0000-0001-5764-0840
- 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
- Received by editor(s): March 6, 2014
- Received by editor(s) in revised form: August 26, 2014
- Published electronically: August 20, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6539-6574
- MSC (2010): Primary 11R42, 19F27
- DOI: https://doi.org/10.1090/tran/6453
- MathSciNet review: 3461042