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Stark's Conjecture and New Stickelberger Phenomena

Published online by Cambridge University Press:  20 November 2018

Victor P. Snaith
Victor P. Snaith*
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, U.K. e-mail: v.snaith@sheffield.ac.uk
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Abstract

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We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic $K$-groups of rings of integers in number fields. Our conjecture ismotivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension when $r\,=\,-2,\,-4,\,-6,\,\ldots $ the Coates–Sinnott conjecture merely predicts that zero annihilates ${{K}_{-2r}}$ of the ring of $S$–integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the étale cohomology of the cyclotomic extensions of the rationals.

Keywords

11G5511R3411R4219F27
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 2 , 01 April 2006 , pp. 419 - 448
Copyright
Copyright © Canadian Mathematical Society 2006

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