Étale cohomology of rank one $\ell$-adic local systems in positive characteristic

Abstract: We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem. Last version: a few typos corrected