On the $\ell$-adic Fourier transform and the determinant of the middle convolution

TL;DR

This paper explores the relationship between the $\, ext{ell}$-adic Fourier transform and the determinant of the middle convolution, providing detailed descriptions of local monodromy and implications for sheaves with quadratic determinants.

Contribution

It offers a detailed analysis of the local monodromy and determinants in the context of $\, ext{ell}$-adic Fourier transforms and middle convolution, extending Katz's work.

Findings

Determinant of sheaves often remains quadratic under middle convolution with quadratic characters.

Provides a detailed description of local monodromy in the tame case.

Connects local $\, ext{epsilon}$-constants with properties of sheaves under convolution.

Abstract

We study the relation of the middle convolution to the $\ell$-adic Fourier transformation in the \'etale context. Using Katz' work and Laumon's theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz' middle convolution functor $\MC_\chi$ in the tame case. The theory of local $\epsilon$-constants then implies that the property of an \'etale sheaf of having an at most quadratic determinant is often preserved under $\MC_\chi$ if $\chi$ is quadratic.