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

Abstract

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 $X$.

27 pages; fixed a mistake in the proof of Proposition 3.3; agrees with the published version in Journal of Algebra

References in corpus (2)

• Higher Topos Theory
• Transformation de Fourier generalisee