p-torsion étale sheaves on the Jacobian of a curve

Suppose X is a smooth, proper, geometrically connected curve over Fq with an Fq-rational point x0. For any Fq×-character σ of π1(X) trivial on x0, we construct functor Lnσ from the derived category coherent sheaves moduli space deformations Witt ring Wn(Fq) to constructible Wn(Fq)-sheaves Jacobian X. The functors categorify Artin reciprocity map for geometric class field theory p-torsion coefficients. We then give criterion fully faithfulness (an enhanced version of) in terms Hasse–Witt matrix