$p$-torsion \'etale sheaves on the Jacobian of a curve

Suppose $X$ is a smooth, proper, geometrically connected curve over $\mathbb F_q$withan$\mathbb F_q$-rationalpoint$x_0$\mathrm{.}Forany$\mathbb F_q^{\times}$-character$\sigma$of$\pi_1(X)$trivialon$x_0$, 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 $W_n(\mathbb F_q)$tothederivedcategoryofconstructible$W_n(\mathbb F_q)$-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 $X$.