# Algebra and Geometry Seminar

Drinfeld's lemma for an F_p-scheme X asserts a close relationship between X and the formal categorical quotient of the product of X with an algebraically closed field divided by the "partial Frobenius" action on the second factor. In particular, Drinfeld shows that these two have the same lisse and constructible etale sheaves.

We describe two different p-adic analogues of this result, both of which have implications for the Langlands correspondence. One is for lisse sheaves on perfectoid spaces and is related to the Fargues-Scholze construction for mixed-characteristic local Langlands; it also gives rise to a multivariate analogue of (phi, Gamma)-module theory for representations of powers of a local Galois group (the latter being joint work with Annie Carter and Gergely Zabradi). The other is for F-isocrystals and D-modules (joint work with Daxin Xu), and is needed for a crystalline analogue of V. Lafforgue's approach to geometric Langlands for reductive groups (parallel to Abe's crystalline analogue of L. Lafforgue's theorem for GL(n)).