Exterior powers in Iwasawa theory

Abstract

The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-$p$ Galois groups with ramification allowed at a maximal set of primes over $p$ such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a $p$-adic $L$-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over $p$ by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by $p$-adic $L$-functions under the relevant CM main conjectures.