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An unconditional main conjecture in Iwasawa theory and applications (2303.13603v1)

Published 23 Mar 2023 in math.NT and math.KT

Abstract: We improve upon the recent keystone result of Dasgupta-Kakde on the $\Bbb Z[G(H/F)]-$-Fitting ideals of certain Selmer modules $Sel_ST(H)-$ associated to an abelian, CM extension $H/F$ of a totally real number field $F$ and use this to compute the $\Bbb Z_p[[G(H_\infty/F)]]-$-Fitting ideal of the Iwasawa module analogues $Sel_ST(H_\infty)_p-$ of these Selmer modules, where $H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$, for an odd prime $p$. Our main Iwasawa theoretic result states that the $\Bbb Z_p[[G(H_\infty/F]]-$-module $Sel_ST(H_\infty)_p-$ is of projective dimension $1$, is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant $p$-adic $L$-function $\Theta_ST(H_\infty/F)$. Further, we establish a perfect duality pairing between $Sel_ST(H_\infty)_p-$ and a certain $\Bbb Z_p[[G(H_\infty/F)]]-$-module $\mathcal M_ST(H_\infty)-$, essentially introduced earlier by Greither-Popescu. As a consequence, we recover the Equivariant Main Conjecture for the Tate module $T_p(\mathcal M_ST(H_\infty))-$, 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 $\Bbb Z[G(H/F)]$-Fitting ideals of the higher Quillen $K$-groups $K_{2n-2}(\mathcal O_{H,S})$, for all $n\geq 2$. 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.

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