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# Stark's Conjecture and New Stickelberger Phenomena

Published:2006-04-01
Printed: Apr 2006
• Victor P. Snaith
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## Abstract

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 is motivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$-adic \'{e}tale cohomology. In addition, the conjecture generalises the well-known Coates--Sinnott conjecture. For example, for a totally real extension when $r = -2, -4, -6, \dotsc$ 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 \'{e}tale cohomology of the cyclotomic extensions of the rationals.
 MSC Classifications: 11G55 - Polylogarithms and relations with $K$-theory 11R34 - Galois cohomology [See also 12Gxx, 19A31] 11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 19F27 - Etale cohomology, higher regulators, zeta and $L$-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10]