We prove a stronger version of the keystone result of Dasgupta and Kakde (Ann Math 197:289–388, 2023) on the \(\mathbb Z[G(H/F)]^-\) -Fitting ideals of certain Selmer modules \(Sel_S^T(H)^-\) associated with 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 \(Sel_S^T(H_\infty )_p^-\) 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 \(Sel_S^T(H_\infty )_p^-\) is of projective dimension 1 (unlike the \({\mathbb {Z}}[G(H/F)]^-\) -module \(Sel_S^T(H)^-\) which could have infinite projective dimension), 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 \(Sel_S^T(H_\infty )_p^-\) and a certain \(\mathbb Z_p[[G(H_\infty /F)]]^-\) -module \({\mathcal {M}}_S^T(H_\infty )^-\) , essentially introduced by Greither and the second author in (J Algebraic Geom 24:629–692, 2015). As a consequence, we recover the Equivariant Main Conjecture for the Tate module \(T_p(\mathcal M_S^T(H_\infty ))^-\) , proved in loc.cit. under the hypothesis that the classical Iwasawa \(\mu \) -invariant associated with H and p vanishes. As a further consequence, we give an unconditional proof of the refined Coates–Sinnott Conjecture, proved in loc.cit. under the same \(\mu =0\) hypothesis, and also proved unconditionally but with different methods by Johnston and Nickel in (An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications, arxiv:2010.03186), 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” (introduced by Wiles (Ann Math (2) 131: 555–565, 1990) and refined by Greither in (Math Z 233: 515–534, 2000)) to strengthen further the keystone result in Dasgupta and Kakde (Ann Math 197:289–388, 2023) and prove, as a consequence, a conjecture of Burns–Kurihara–Sano on Fitting ideals of Selmer groups of CM number fields.
