Understanding the structures and symmetries in chromatic homotopy theory, such as Picard and Brauer groups, is essential for exploring the foundations of stable homotopy theory. While the K(n)-local Picard, Brauer, and Galois groups have been extensively studied, much less is known about their T(n)-local counterparts.
Addressing this gap, we construct new telescopic Picard and Brauer elements and connect these to Galois theory. Using this we lift a non-abelian Galois extension of the K(n)-local sphere to the telescopic setting.
Time permitting, we will discuss higher Galois extensions and an extended Kummer theory linking these with higher Brauer groups.