2025/03/29 by Gambheera, Rusiru, Popescu, Cristian D.
#11R23(Primary) 11R29 #11R34 #11R42 (Secondary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2503.23192
We consider a finite, abelian, CM extension H/F of a totally real number field F, and construct a ℤp[[G(H_∞/F)]]-module ∇ST(H_∞)p, where p>2 is a prime and H_∞ is the cyclotomic \Bbb Zp-extension of H. This is the Iwasawa theoretic analogue of a module introduced by Ritter and Weiss in \citeRitter-Weiss and studied further by Dasgupta and Kakde in \citeDasgupta-Kakde. Our main result states that the \Bbb Zp[[G(H_∞/F]]--module ∇ST(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). As a first application, we compute the Fitting ideal of an arithmetically interesting \Bbb Zp[[G(H_∞/F)]]--module XST,-, which is a variant of the classical unramified Iwasawa module X (the Galois group of the maximal abelian, unramified, pro-p extension of H_∞), extending earlier results of Greither-Kataoka-Kurihara \citeGreither-Kataoka-Kurihara. These are all instances of what is now called an Equivariant Main Conjecture in the Iwasawa theory of totally real number fields, and refine the classical main conjecture, proved by Wiles in \citewiles. As a final application, we give a short, Iwasawa theoretic proof of the minus p-part of the far-reaching Equivariant Tamagawa Number Conjecture for the Artin motive hH/F, for all primes p>2, a result also obtained, independently and with different (Euler system) methods, by Bullack-Burns-Daoud-Seo \citeBullach-Burns-Daoud-Seo and Dasgupta-Kakde-Silliman \citeDasgupta-Kakde-Silliman-ETNC.