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

@article{Esnault2019taleCO,
title={{\'E}tale cohomology of rank one \$\$\ell \$\$-adic local systems in positive characteristic},
author={H'elene Esnault and Moritz Kerz},
journal={arXiv: Algebraic Geometry},
year={2019}
}
• Published 22 August 2019
• Mathematics
• arXiv: Algebraic Geometry
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.
3 Citations
Density of Arithmetic Representations of Function Fields
• Mathematics
Épijournal de Géométrie Algébrique
• 2022
We propose a conjecture on the density of arithmetic points in the deformation space of representations of the \'etale fundamental group in positive characteristic. This? conjecture has applications
Level structure, arithmetic representations, and noncommutative Siegel linearization
• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2022
Abstract Let ℓ{\ell} be a prime, k a finitely generated field of characteristic different from ℓ{\ell}, and X a smooth geometrically connected curve over k. Say a semisimple representation of

## References

SHOWING 1-10 OF 40 REFERENCES
Arithmetic subspaces of moduli spaces of rank one local systems
• Mathematics
Cambridge Journal of Mathematics
• 2020
We show that closed subsets of the character variety of a complex variety with negatively weighted homology, which are $p$-adically integral and Galois invariant, are motivic. Final version:
Cohomology jump loci of quasi-projective varieties
• Mathematics
• 2012
We prove that the cohomology jump loci in the space of rank one local systems over a smooth quasi-projective variety are finite unions of torsion translates of subtori. The main ingredients are a
Polarizable twistor D-modules
We prove a Decomposition Theorem for the direct image of an irreducible local system on a smooth complex projective variety under a morphism with values in another smooth complex projective variety.
Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville
• Mathematics
• 1987
Introduction 389 w Notation and conventions 392 w Deformations of cohomology groups 392 w Generic vanishing criteria for topologically trivial line bundles 397 w A Nakano-type generic vanishing
Subspaces of moduli spaces of rank one local systems
Suppose X is a smooth projective variety. The moduli space M (X) of rank one local systems on X has three different structures of complex algebraic group (Betti, de Rham, and Dolbeault). A subgroup
Vanishing theorems for perverse sheaves on abelian varieties, revisited
• Mathematics
• 2017
We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new
Vanishing theorems for constructible sheaves on abelian varieties
• Mathematics
• 2015
We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for
Higher obstructions to deforming cohomology groups of line bundles
• Mathematics
• 1991
Our purpose is to study the cohomological properties of topologically trivial holomorphic line bundles on a compact Kahler manifold. (See [GLl, GL2] for our prior work on this topic.) Let M be a
Rigid analytic geometry and its applications
• Mathematics
• 2003
Preface.- Valued fields and normed spaces.- The projective line.- Affinoid algebras.- Rigid spaces.- Curves and their reductions.- Abelian varieties.- Points of rigid spaces, rigid cohomology.- Etale
Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor D-Modules
Part 4. An Application to the theory of Pure Twistor $D$-modules: Pure twistor $D$-module Prolongation of $\mathcal R$-module $\mathcal E$ The filtrations of $\mathfrak{E} [\eth_t]$ The weight