Iwasawa invariants of finite spectra

TL;DR

This paper computes Iwasawa invariants for modules related to the $p$-adic $K$-theory of finite spectra, revealing growth patterns and confirming the vanishing of $$-invariants, and establishes a spectral analogue of the Iwasawa Main Conjecture.

Contribution

It introduces explicit calculations of Iwasawa invariants for $K$-theory modules of finite spectra and proves a spectral analogue of the Iwasawa Main Conjecture.

Findings

The average order growth is asymptotically   imes (-_p(n))

All Iwasawa -invariants of finite spectra are zero

A spectral analogue of the Iwasawa Main Conjecture is established

Abstract

We calculate the classical Iwasawa invariants of the Iwasawa modules associated to the $p$-adic topological $K$-theory of finite spectra. We show that the graded average of the orders of $n$ consecutive $K(1)$-local homotopy groups of a finite spectrum $X$ grows asymptotically like $\frac{-\log_p(n)}{2}$ times the total Iwasawa $\lambda$-invariant of $X$. We show that the Iwasawa $\mu$-invariants of finite spectra are all zero. Finally, we prove a spectral analogue of a weak form of the Iwasawa Main Conjecture, describing the orders of the $K(1)$-local homotopy groups of a certain ``torsion-free replacement'' of $X$ in terms of the characteristic polynomials of the Iwasawa modules associated to $X$.