Geometric local epsilon factors

TL;DR

This paper provides an explicit cohomological definition of geometric local epsilon factors for dic Galois representations over henselian discrete valuation fields of positive characteristic, extending classical notions with new explicit formulas and properties.

Contribution

It introduces a cohomological framework for local epsilon factors in positive characteristic, with explicit formulas and compatibility properties, generalizing previous work by Laumon and Deligne.

Findings

Defined geometric local dic epsilon factors explicitly

Established induction and compatibility properties

Proved a product formula for dic sheaves on curves

Abstract

Inspired by the work of Laumon on $\varepsilon$-factors and by Deligne's $1974$ letter to Serre, we give an explicit cohomological definition of $\varepsilon$-factors for $\ell$-adic Galois representations over henselian discrete valuation fields of positive equicharacteristic $p \neq \ell$, with (not necessarily finite) perfect residue fields. These geometric local $\varepsilon$-factors are completely characterized by an explicit list of purely local properties, such as an induction formula and the compatibility with geometric class field theory in rank $1$, and satisfy a product formula for $\ell$-adic sheaves on a curve over a perfect field of characteristic $p$.