Stark's Conjecture and New Stickelberger Phenomena

Abstract

AbstractWe introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraicK–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 tol–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension whenr= –2,–4,–6, … the Coates–Sinnott conjecturemerely predicts that zero annihilatesK–2rof the ring ofS–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.