On The Telescopic Picard Group

We prove that for any prime $p$ and height $n\ge 1$, the telescopic Picard group $\mathrm{P}\mathrm{i}\mathrm{c}({\mathrm{S}\mathrm{p}}_{Tn})$ contains a subgroup of the form ${\mathbb{Z}}_p\times \mathbb{Z}\mathrm{/}{a}_p(p^n-1)$, where $a_p=1$ if $p=2$ and $a_p=2$ if $p$ is odd. Using Kummer theory, we obtain an $({\mathbb{F}}_{p^n}^{\times }⋊\mathbb{Z}\mathrm{/}n)$-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.