An unconditional main conjecture in Iwasawa theory and applications

Abstract

We improve upon the recent keystone result of Dasgupta-Kakde on the $\mathbb{Z}[G(H/F){]}^{-}$-Fitting ideals of certain Selmer modules $Se{l}_S^T(H{)}^{-}$ associated to an abelian, CM extension $H/F$ of a totally real number field $F$ and use this to compute the ${\mathbb{Z}}_p[[G(H_{\infty }/F)]{]}^{-}$-Fitting ideal of the Iwasawa module analogues $Se{l}_S^T(H_{\infty }{)}_p^{-}$ of these Selmer modules, where $H_{\infty }$ is the cyclotomic ${\mathbb{Z}}_p$-extension of $H$, for an odd prime $p$. Our main Iwasawa theoretic result states that the ${\mathbb{Z}}_p[[G(H_{\infty }/F]{]}^{-}$-module $Se{l}_S^T(H_{\infty }{)}_p^{-}$ is of projective dimension $1$, is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant $p$-adic $L$-function ${\Theta }_S^T(H_{\infty }/F)$. Further, we establish a perfect duality pairing between $Se{l}_S^T(H_{\infty }{)}_p^{-}$ and a certain ${\mathbb{Z}}_p[[G(H_{\infty }/F)]{]}^{-}$-module ${\mathcal{M}}_S^T(H_{\infty }{)}^{-}$, essentially introduced earlier by Greither-Popescu. As a consequence, we recover the Equivariant Main Conjecture for the Tate module $T_p({\mathcal{M}}_S^T(H_{\infty }){)}^{-}$, proved by Greither-Popescu under the hypothesis that the classical Iwasawa $\mu$-invariant associated to $H$ and $p$ vanishes. As a further consequence, we give an unconditional proof of the refined Coates-Sinnott Conjecture, proved by Greither-Popescu under the same $\mu =0$ hypothesis, and also recently proved unconditionally but with different methods by Johnston-Nickel, regarding the $\mathbb{Z}[G(H/F)]$-Fitting ideals of the higher Quillen $K$-groups $K_{2n-2}({\mathcal{O}}_{H,S})$, for all $n\ge 2$. Finally, we combine the techniques developed in the process with the method of ''Taylor-Wiles primes'' to strengthen further the keystone result of Dasgupta-Kakde and prove, as a consequence, a conjecture of Burns-Kurihara-Sano on Fitting ideals of Selmer groups of CM number fields.