Drinfeld's lemma for schemes

In the course of constructing the Langlands correspondence for GL(2) over a function field, Drinfeld discovered a surprising fact about the interaction between étale fundamental groups and products of schemes in characteristic p. We state this result, describe a new approach to it involving a generalization to perfectoid spaces, and mention an application in p-adic Hodge theory (from joint work with Carter and Zabradi).