An Equivariant Main Conjecture in Iwasawa theory and applications

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 and studied extensively from a Galois module structure point of view in our previous works. 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. 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).