Matthew Emerton
An important result in the theoy of l-adic etale sheaves on varieties in characteristic p is Grothendieck's cohomological formula for the L-function of such a sheaf. N. Katz conjectured that an analogous formula should hold for p-adic etale sheaves, although in the p-adic setting, there is no framework of Poincare duality or the Lefschetz trace formula available to prove it. In this talk we will present a proof of Katz's formula, due to the speaker and M. Kisin. The key technique of the proof is to exploit the connection between p-adic sheaves and F-crystals, which allows one to reduce calculations of etale cohomology to more accesible coherent cohomological calculations.