Iwasawa invariants of finite spectra

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 $X$.