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.
Categories: math.AT math.CT