The algebra Zℓ[[Zpd]] and applications to Iwasawa theory

Let $\ell$ and $p$ be distinct primes, and let $\Gamma$ be an abelian pro-$p$-group. We study the structure of the algebra $\Lambda:=\mathbb{Z}_\ell[[\Gamma]]$ and of $\Lambda$-modules. The algebra $\Lambda$ turns out to be a direct product of copies of rings of integers of unramified cyclotomic extensions of $\mathbb{Q}_\ell$ and this induces a similar decomposition for a family of $\Lambda$-modules. Inside this family we define Sinnott modules and provide characteristic ideals and formulas \`a la Iwasawa for orders and ranks of their quotients. When $\Gamma\simeq \mathbb{Z}_p^d$\, 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 $\mathbb{Z}_p$-extensions. Moreover, for global function fields of positive characteristic we use the specialization of a Stickelberger series to define an element in $\Lambda$ 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 $\mathbb{Z}_p$-extensions.