The mod p cohomology functors K(n) admit the unit groups D^x of certain p-adic division algebras as `Adams' (ie stable multiplicative) automorphism. I argue that these functors lift to cohomology theories indexed by maximal toruses $L^x \subset D^x$, with values in modules over the valuation rings of $L$. A theorem of Weil and Shafarevich identifies the normalizers of these toruses with the Galois groups of certain extensions $L^\infty/ \mathbb{Q}_p,$ suggesting that these lifted functors resemble some kind of algebraic K-theory of (the perfectoid completion) of $L^\infty.$

[These abelian extensions are analogous to the infinite cyclic extensions of link complements, and are maybe not as unfamiliar to topologists as they might look.]

Refs: arXiv:1808.08587, arXiv:1608.04702