2023/03/23 by Rusiru Gambheera, Gambheera, Rusiru, Cristian D. Popescu +1
Mathematics · #11R23 #11R29 #11R34 #11R42 #11R70 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2303.13603
openalex publication_date 2023/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We improve upon the recent keystone result of Dasgupta-Kakde on the \Bbb Z[G(H/F)]--Fitting ideals of certain Selmer modules SelST(H)- associated to an abelian, CM extension H/F of a totally real number field F and use this to compute the \Bbb Zp[[G(H_∞/F)]]--Fitting ideal of the Iwasawa module analogues SelST(H_∞)p- of these Selmer modules, where H_∞ is the cyclotomic \Bbb Zp-extension of H, for an odd prime p. Our main Iwasawa theoretic result states that the \Bbb Zp[[G(H_∞/F]]--module SelST(H_∞)p- is of projective dimension 1, is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant p-adic L-function ΘST(H_∞/F). Further, we establish a perfect duality pairing between SelST(H_∞)p- and a certain \Bbb Zp[[G(H_∞/F)]]--module \mathcal MST(H_∞)-, essentially introduced earlier by Greither-Popescu. As a consequence, we recover the Equivariant Main Conjecture for the Tate module Tp(\mathcal MST(H_∞))-, proved by Greither-Popescu under the hypothesis that the classical Iwasawa μ-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 μ=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 K2n-2(\mathcal OH,S), for all n≥ 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.