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

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.