K-local homotopy of symmetric product spectra

Compute explicitly the homotopy groups of the K(k+1)-localization L_{K(k+1)}\mathbb{Sp}^{p^k}, in a form analogous to the known computation for k=0.

Background

The symmetric product spectrum \mathbb{Sp}{pk} is K(i)-acyclic for 0<i\leq k and has controlled K(k+1)-cohomology. The paper identifies the first chromatic height at which it is nontrivial but does not compute its localized homotopy groups.

References

For $k > 0$, in the range of convergence $0 < * < c(n,k+1)-n$, one has

\pi_{n+}Sn \cong \frac{\pi_\hat Sn_k}{\pi_{*+2k-1}\mathbb{Sp}{pk}}

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 5, subsection “Survival of elements from S^1_k”

Can we give an explicit computation of $\pi_*L_{K(k+1)}\mathbb{Sp}{pk}$, analogous to the known case $k = 0$?

— An abelian model for the Goodwillie tower of the circle  (2609.31269 - Nervo, 25 Sep 2026) in Section 2, subsection “Symmetric products”