Number Theory Seminar

The proof by L. Lafforgue of the Langlands correspondence for $\text{GL}_n$ over a function field over a finite field implies, among other things, that "all" l-adic etale sheaves on a curve (more precisely, all irreducible sheaves with determinant of finite order) are "of motivic origin." Subsequent advances in p-adic cohomology enabled Abe to port Lafforgue's proof to the category of overconvergent $F$-isocrystals. This can be exploited to gain some consequences on both the l-adic and p-adic sides. I will survey a few of these, including a joint result with Drinfeld on Newton polygons of l-adic sheaves, and a recent result of Abe-Esnault giving a sort of converse of Deligne's conjecture on crystalline companions (which itself remains open).