After an introduction into the classical Weil conjectures, I will show how one can formulate and prove an analogue of the Weil Conjectures for the moduli stack of vector bundles on an algebraic curve over a field in positive characteristic. Ingredients will include a complete description of the l-adic cohomology ring of the moduli stack in terms of Atiyah-Bott characteristic classes and the actions of the various Frobenius morphisms on it. The geometry of some of these actions is still very mysterious. If time permits I will end by suggesting how all these calculations might actually be governed by an etale homotopy type a la Artin-Mazur which appears implicit also in recent work by Gaitsgory and Lurie.

海报