Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

The term "Drinfeld's lemma" refers to several statements that relate the geometry of a scheme over Fp to the result of base-extending to an algebraically closed field, then formally dividing by the "partial Frobenius" action on the field (fixing the original scheme). One such statement asserts that for any prime 6 = p, lisse-adic sheaves on the original scheme are the same as on this formal quotient. This result plays a pivotal role in the approach to the Langlands correspondence for a reductive group over the function field of a curve over a finite field, pioneered by Drinfeld for the group GL(2) and subsequently extended by L. Lafforgue and then V. Lafforgue. In this lecture, we describe an analogue of this statement involving "lisse p-adic sheaves", meaning overconvergent F -isocrystals. The hope is that this can be used to upgrade Abe's proof of the Langlands correspondence for GL(n) (based on L. Lafforgue's method) to more general reductive groups. Since overconvergent F -isocrystals are not directly described by representations of the profi- nite Žetale fundamental group, the p-adic statement is not an immediate corollary of the original Drinfeld's lemma. We deduce it by building up some structural properties of "?-isocrystals" (iso- crystals with multiple Frobenius structures), particularly the Newton polygon variation and slope filtration, so that we can eventually reduce to the prior result.