The algebra Z_l[[Z_p^d]] and applications to Iwasawa theory

Let 𝓁 and 𝑝 be distinct primes, and let Γ be an abelian pro-𝑝-group. We study the structure of the algebra Λ∶= ℤ𝓁[[Γ]] and of Λ-modules. The algebra Λ turns out to be a direct product of copies of rings of integers of unramified cyclotomic extensions of ℚ_𝓁, and this induces a similar decomposition for a family of Λ 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 Γ≃ℤ^𝑑_𝑝 is the Galois group of an extension of global fields, 𝓁-class groups and (duals of) 𝓁-Selmer groups provide examples of Sinnott modules, and our formulas vastly extend results of Washington and Sinnott on 𝓁-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 Λ which interpolates special values of Artin 𝐿-functions. With this element and the characteristic ideal of 𝓁 class groups, we formulate an Iwasawa main conjecture (IMC) for this setting and prove some special cases of it for relevant ℤ_𝑝-extensions.