We prove a version of the quantum geometric Langlands conjecture in characteristic p. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline D-modules on the stack of rank N vector bundles on an algebraic curve C in characteristic p. The twisting parameters are related in the way predicted by the conjecture and are assumed to be irrational (i.e., not in F-p). We thus extend some previous results Braverman and Bezrukavnikov concerning a similar problem for the usual (nonquantum) geometric Langlands. In the course of the proof we introduce a generalization of p-curvature for line bundles with nonflat connections, define quantum analogues of Hecke functors in characteristic p, and construct a Liouville vector field on the space of de Rham local systems on C.
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