On The Telescopic Picard Group

Abstract

We prove that for any prime $p$ and height $n \ge 1$, the telescopic Picard group $\mathrm{Pic}(\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\mathbb{Z}_p \times \mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$ is odd. Using Kummer theory, we obtain an $(\mathbb{F}_{p^n}^\times \rtimes \mathbb{Z}/n)$-Galois extension of $\mathbb{S}_{T(n)}$, obtaining the first example of a lift of a non-Abelian Galois extension of the $K(n)$-local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework for the periodicity theorem, utilizing the symmetries of this framework to construct Picard elements.