The conjectures in the title were formulated in the late 1970's as vast generalizations of the classical theorem of Stickelberger. They make a subtle connection between the Z[G(L/k)]-module structure of the Quillen K-groups K*(OL) in an abelian extension L/k of number fields and the values at negative integers of the associated G(L/k)-equivariant L-functions.

These conjectures are known to hold true if the base field k is Q, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields k.