K-theory and periodicity

The chromatic perspective organizes elements in p-local homotopy groups into periodic families whose period depends on a height n. A simplified form of the redshift conjecture of Ausoni and Rognes asks if algebraic K-theory increases height by exactly one. In joint work with J. Hahn and D. Wilson, we prove this in the case of the prime skew fields in homotopy theory known as Morava K-theory. In my talk, I will mostly spend time motivating this result and perhaps mention a new tool from arithmetic geometry that is key to the proof.