Asymptotics of coinvariants of Iwasawa modules under non-normal subgroups

Let G be a pro-p p-adic analytic group, thought of as a closed subgroup of GLN (Zp), and let Σ be a closed subgroup of G. Write Λ for the completed group algebra Zp[[G]] and let M be a finitely generated Λ-module. Let G = G ⊃ G ⊃ G ⊃ . . . be the descending sequence of principal congruence subgroups of G; write Gn for the quotient G/G and Σn for the image of Σ in Gn. Write Mn for the coinvariant quotient of M under G. Then Mn is a module for the group algebra Zp[Gn]. In Iwasawa theory, one often finds that the growth of arithmetic invariants of interest (e.g. class numbers, Mordell-Weil ranks) is controlled by a Λ-module M . In particular, the growth can be related to the Zp-ranks of the coinvariant quotients ofM by various subgroups of G. Understanding these quotients is a purely algebraic problem. For instance, Harris [3, Theorem 1.10] shows that, if M is a Λ-torsion module, rankZp Mn = O(p n(dimG−1)). (1)