# Exterior powers in Iwasawa theory

## TL;DR

This paper explores advanced Iwasawa modules related to CM fields, examining their structure through exterior powers and p-adic L-functions, extending classical conjectures to more general modules.

## Contribution

It introduces a framework for analyzing smaller Iwasawa modules via exterior powers and relates their support to p-adic L-functions under CM main conjectures.

## Key findings

- Higher codimension support measured by p-adic L-functions
- Generalization of main conjectures to exterior power quotients
- Connections between inertia subgroups and Iwasawa module structure

## 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 Iwasawa 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 p-adic L-functions under the relevant CM main conjectures.