Local Fourier transform and epsilon factors

TL;DR

This paper explicitly computes the local Fourier transform of certain monomial representations over local fields and establishes a formula linking epsilon factors to the determinant of this transform, advancing understanding of $ ext{ell}$-adic sheaves.

Contribution

It provides explicit calculations of the local Fourier transform for monomial representations and proves a formula relating epsilon factors to the transform's determinant.

Findings

Explicit formulas for local Fourier transforms of monomial representations

A new relation between epsilon factors and the determinant of the Fourier transform

Enhanced tools for studying $ ext{ell}$-adic sheaves and Galois representations

Abstract

Laumon introduced the local Fourier transform for $\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\ell$, as a powerful tool to study the Fourier-Deligne transform of $\ell$-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon's formula relating the epsilon factor to the determinant of the local Fourier transform under the same condition.