$p$-torsion étale sheaves on the Jacobian of a curve

Suppose $X$ is a smooth, proper, geometrically connected curve over $\mathbb F_q$ with an $\mathbb F_q$-rational point $x_0$. For any $\mathbb F_q^{\times}$-character $\sigma$ of $\pi_1(X)$ trivial on $x_0$, we construct a functor $\mathbb L_n^{\sigma}$ from the derived category of coherent sheaves on the moduli space of deformations of $\sigma$ over the Witt ring $W_n(\mathbb F_q)$ to the derived category of constructible $W_n(\mathbb F_q)$-sheaves on the Jacobian of $X$. The functors $\mathbb L_n^{\sigma}$ categorify the Artin reciprocity map for geometric class field theory with $p$-torsion coefficients. We then give a criterion for the fully faithfulness of (an enhanced version of) $\mathbb L_n^{\sigma}$ in terms of the Hasse-Witt matrix of $X$.