The algebra $\mathbb{Z}_\ell[[\mathbb{Z}_p^d]]$ and applications to Iwasawa theory

TL;DR

The paper develops a comprehensive algebraic framework for the ℓ-adic Iwasawa algebra Λ = Z_ℓ[[Γ]] with Γ an abelian pro-p group, proving a decomposition Λ ≅ ∏_[ω] Z_ℓ[ω(Γ)] and introducing Sinnott modules whose Λ_[ω]-components admit explicit rank and torsion formulas. It applies this theory to ℓ-class groups and ℓ-Selmer groups in Z_p^d-extensions, establishing p-adic convergence results and generalizing Washington and Sinnott-type bounds in a uniform Λ-analytic setting. A Stickelberger-based interpolation in function field contexts is developed to formulate an Iwasawa Main Conjecture and partial results are proved for weak IMC in p-cyclotomic and arithmetic Z_p-extensions, with explicit Ω and S factors controlling p-adic valuations. Overall, the work provides a unified algebraic approach to ℓ-parts of Iwasawa theory, enabling precise growth formulas, normic-system constructions, and a pathway toward Main Conjectures across both number field and function field settings. Its framework paves the way for further χ-version IMCs and potential extensions to non-abelian contexts through the Λ-structure and componentwise analysis.

Abstract

Let $\ell$ and $p$ be distinct primes, and let $\G$ be an abelian pro-$p$-group. We study the structure of the algebra $Ł:=\Z_\ell[[\G]]$ and of $Ł$-modules. The algebra $Ł$ turns out to be a direct product of copies of ring of integers of cyclotomic extensions of $\Q_\ell$ and this induces a similar decomposition for a family of $Ł$-modules. Inside this family we define Sinnott modules and provide characteristic ideals and formulas à la Iwasawa for orders and ranks of their quotients. When $\G\simeq \Z_p^d$\, is the Galois group of an extension of global fields, $\ell$-class groups and (duals of) $\ell$-Selmer groups provide examples of Sinnott modules and our formulas vastly extend results of L. Washington and W. Sinnott on $\ell$-class groups in $\Z_p$-extensions. Moreover, for global function fields of positive characteristic we use the specialization of a Stickelberger series to define an element in $Ł$ which interpolates special values of Artin $L$-functions. With this element and the characteristic ideal of $\ell$-class groups we formulate an Iwasawa Main Conjecture for this setting and prove some special cases of it for relevant $\Z_p$-extensions.

Theorems & Definitions (72)

• Theorem 1.1: Theorem \ref{['t:struttura1']}
• Theorem 1.2
• Remark 1.3
• Theorem 1.4: Corollary \ref{['co:IMCZpArith']}
• Theorem 2.1
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• proof