  We construct a new class of Iwasawa modules, which are the number field
analogues of the p-adic realizations of the Picard 1-motives constructed by
Deligne in the 1970s and studied extensively from a Galois module structure
point of view in our recent work. We prove that the new Iwasawa modules are of
projective dimension 1 over the appropriate profinite group rings. In the
abelian case, we prove an Equivariant Main Conjecture, identifying the first
Fitting ideal of the Iwasawa module in question over the appropriate profinite
group ring with the principal ideal generated by a certain equivariant p-adic
L-function. This is an integral, equivariant refinement of the classical Main
Conjecture over totally real number fields proved by Wiles in 1990. Finally, we
use these results and Iwasawa co-descent to prove refinements of the
(imprimitive) Brumer-Stark Conjecture and the Coates-Sinnott Conjecture, away
from their 2-primary components, in the most general number field setting. All
of the above is achieved under the assumption that the relevant prime p is odd
and that the appropriate classical Iwasawa mu-invariants vanish (as conjectured
by Iwasawa.)

Number Theory